Long answer warning: If you wish to skip all the context, just see last section about Mangalloy steel.
The Problems with Tungsten
While tungsten is an incredibly hard metal often used in making tools designed to cut steel, it is not actually as tough as most grades of steel. It is also a much heavier metal than iron such that most weapons made of tungsten would be inherently too fragile and too heavy to be as good as steel. Its easiness to shatter would also make it a very bad armor material compared to steel. It's melting point is also so high that it would take a magic forge to even work with.
The only practical application of tungsten as a weapon making material would perhaps be in the formation of bimetallic blades made of tungsten carbide encased in a mild steel (like how the katana is made from a harder steel encased in a mild steel). This would in theory make a decent blade, but if you already allow for unlimited cost, magic smelting techniques, and metallurgical knowledge, then a better option for tungsten is to use a blue steel. Blue steel is a high end iron alloy made with small amounts of tungsten and chromium which is used in premium knife and sword making today for its combination of excellent edge retention, weight, and toughness.
The Problems with Titanium
Pure titanium is a much softer and all around weaker metal than steel. Much like pure iron, it is too soft, but unlike iron, it can't be easily alloyed with carbon until it reaches its desired hardness... or rather, it takes so little carbon to harden titanium that it very quickly becomes too hard, brittle, and useless; so, you cant make it with traditional forging techniques at all. For titanium to be at all useful, you will need magic to refine it, and you will also need other anachronistic metals like Aluminum and Vanadium. If you allow for all of this, titanium alloy CAN make a very good armor. It's weaker than most steels of a given thickness, but for its weight it is generally stronger. So, if you want a light weight armor that can stop most attacks, titanium alloy is a good choice.
However, it is not a good blade making material. Even a basic carbon steel makes for better blades than titanium alloy because, titanium can't hold a narrow edge as well. In order to make an equally tough blade out of titanium, it would need to be thicker, and thus not cut as well.
That said, titanium is still not the BEST material to make body armor out of. There are types of steel that are even stronger for their weight than any titanium alloy, and their higher density means that you could make a thinner suit of lightweight armor than a titanium alloy would allow which would not just offer great protection, but the thinner plates/smaller rings will allow better flexibility in your articulating sections.
Solution: Steel, but a better explanation than existing answers
The first important thing to understand is that steel is not one single alloy, but rather a general term meaning any iron based alloy... but all iron is an iron alloy, because it is practically impossible to make a truly pure iron. Some of the stuff we call iron actually has a lower iron content than the stuff we call steel and vise versa. Some of the stuff we call iron is even chemically identical to other stuff we call steel. In short, differentiating iron and steel in the modern context is pretty pointless.
The best way to understand what steel is is to look at the word's historical usage. The word steel is derived from the Old English word stēli which simply means "stay". In the medieval period, blacksmiths would often advertise their prices for "blaec" meaning "black" which we now call iron or "stēli blaec" meaning "stayed black" which later became "steeled iron" and then shortened to just steel. Simply put, steel or steeled iron just means that the blacksmith did something extra to it to make it "stay" or be stronger than other iron.
The second important thing to understand is that by the end of the Medieval Period, steel making was actually very good. Expert blacksmiths could make steels that were very comparable modern steel. No they weren't as well alloyed or refined as modern steel, but complex wroughting and tempering practices gave their steels the precisely desired qualities in the exact places where it mattered most in a way that you don't get from modern homogenized steels.
So, depending on what part of the medieval period you are talking about, the kind of "super metals" you are talking about already existed. A high quality, late medieval weapon/armor grade steel used by knights and kings was way more rare and expensive than the much cheaper steels used in peasant tools and levee weapons, and they could achieve all of the desired effects that you are asking for. To understand this, let's look at all the factors that determine how good your steel really is.
- Contamination: Throughout ancient history, sulfur and phosphorous contamination were major issues in the iron industry. These contaminants would make iron alloys significantly more brittle without adding significant hardness; so, it was common for blades to shatter if they were made from a contaminated ore. During the early parts of the medieval period, many blacksmiths learned to cook their ore at a lower temperature before they would smelt it. This burned away most of the contamination and lead to a much more pure steel. So, if your setting represents the early medieval period, a smith could add to the price, quality, and rarity of his steel just by pre-cooking his ore. This step was often skipped when making budget steel or tools where shattering was not a major concern. It was also not commonly done in Middle-Eastern blacksmithing; so, depending on where you are, decontaminated steel could be rare. So, a cheap levee's blade might shatter, even in the late medieval period whereas a knight's blade would be far less likely to be contaminated.
- Carbon Content: When smelting iron ore, the iron always absorbs a certain amount of carbon from the kiln. The more carbon it has, the harder and more brittle your steel. Then as you hammer your steel into shape, you press out some of the carbon making it softer. The ideal weapon/armor grade steel is a medium carbon alloy (about .3% to .5% percent depending on what you are making). The problem most early blacksmiths ran into was that they had to be careful not to add too much carbon to their steel or it would crumble when they would try to hammer it out, or they would over hammer it making it too soft. So getting a good, medium carbon steel was almost impossible. By the high medieval period, you start to see finery forges that could take higher carbon steels and reduce the carbon until it was ideal for forging. It took a LOT of extra charcoal, time, and equipment to make finery steel. In fact, finery steel was about 10 times as expensive as traditional bloomery steel in many places where you saw them both as options, but it made a big difference between being able to make the short swords and chain armor of the early medieval period and the longer swords and partial plate armor of the high medieval period.
- Tempering: The way you cool your steel also impacts how good it is. If you cool steel too slowly it anneals meaning that it will bend if you flex it and then stay bent. If you cool it too quickly then it hardens meaning that it will be harder but break if you flex it. If you cool it at just the right speed for the amount of carbon in it, it tempers meaning that if you flex it, it will spring back into its original shape. A good temper can make any blade or armor several times as difficult to damage. In the early-to-high medieval period most blades were either made of a lower carbon steel that was quenched in water, medium carbon 1/2 tempered steel which is where you used a higher carbon steel and water quench just the edge letting the core slow cool, or higher carbon air cooled steel. In the late medieval period, smiths got a lot better at tempering medium carbon steels by using oil instead of water. This was what made it possible to fully replace chain armor with articulating plate armor because it meant that moderate impacts would not dent the armor and lock up your joints. It's also what made longswords and rapiers possible because you could make a long thin blade that could flex and spring back into shape without breaking. When you combine oil tempering, pre-cooking, and finery forging and all the mastery it takes to do those things right, you get a very expensive steel that was practically indestructible when used as armor or weapons in melee combat.
There were of course other trends that came and went like pattern welding, bi-metallic layering, and Damascus alloys which were different ways of alternating high and low carbon steels to make a ridged blade that is both hard and tough, but these methods tend to suffer from delamination under too many impacts and largely fell out of use by the mid-to-late medieval period when homogenized steels got good enough that the trade-off stopped being worth while.
There was also a huge difference in sharpening techniques. For example, Eastern swords were often made out of worse steels than Western swords, but their culture treated sharpening and polishing as its own specialized profession; so, their swords tended to be much sharper. This leads to the general perception that Eastern steel was better, even though they were actually centuries behind in metallurgy.
The actual best metal: Mangalloy Steel
Okay, so what if you wanted to pull out all of the stops, allow magic and modern alloying, and just make the best possible metal. What do you do?
Well, the answer is still steel, but it is a very specific type of steel called Mangalloy Steel. The addition of Manganese to steel is the biggest difference between modern steel and most historical steels. The ancient Spartans technically made their own version of Manganese steel which contributed significantly to their military superiority, but they never shared the secrete of their steel, and its use died out with their civilization, not to be rediscovered until MUCH later. But Mangalloy Steel is very special, even among steels. Most modern steels contain 0.15 to 0.8% manganese which makes it both harder and tougher than carbon steels, but still keeping the steel soft and ductile enough to be grinded, shaped, and generally worked into its final needed form. More than 0.8% manganese ruins your steel... that is until you add enough that something neat happens.
Mangalloy Steel has about 12% Manganese. At this concentration, the steel stops becoming brittle, and instead becomes a work-hardening alloy meaning that the more you try to shape it, the harder and more difficult to shape it becomes as opposed to other steels that loose hardness as you work it. However, everything about it makes it almost impossible to make anything but very basic shapes out of, even using the best of modern technology. Once work hardened, it becomes about 5-10x as strong as most other steels making it even stronger for its weight that titanium, and able to be used to make extremally sharp and durable edges and amazingly durable armor... except that it can't be tooled to make particularly sharp edges or complex armor.
This is where magic comes in. If you could magically shape and sharpen Mangalloy steel, then you could make highly durable, razor sharp swords that exceed even our modern blade making abilities, and complex armor that is thinner and lighter than historical plate armor, while also being several times stronger.
So your rare super metal is actually Manganese, but it has to be mixed with iron to achieve the desired result.