To answer your three main questions:
Could these nations have to come back to medieval style combat with swords, hammers, spears?
Probably not. While it would obviously change how people fight, it probably wouldn't result in a reversion to classical melee weapons being the preferred option. There's too much advantage to killing someone from a distance, and based on what you've said flamethrowers and other chemical dispersion weapons would probably still work.
Could an armour that stops small, high velocity projectiles be vulnerable against melee weapons?
Yes, most likely it would. In fact, modern military body-armor should in theory be very vulnerable to attacks with bludgeoning weapons like maces or warhammers, and most anti-ballistic vests can be cut pretty easily, but these facts are often not taken advantage of because it's much safer to engage a target who is equipped with a firearm from a distance than it is at close range.
Could arrow-like projectiles (slow but heavier) still be able to pierce that armour?
That depends on how the armor derives its invulnerability. In a realistic scenario using actual physics, faster projectiles penetrate better than slower ones under otherwise identical conditions. Velocity isn't everything though, and energy matters too (this is why .45 ACP, while much slower than 9x19mm, generally has better penetration, it's got about 20% more energy on impact than the 9x19mm).
With a 50 pound draw, you can expect a typical carbon-fiber arrow to be moving at about 110 m/s as it leaves the bow and carrying about 147 J of energy. That's not very fast, and it's a miniscule amount of energy (about 35 calories or about 40 milliwatt hours). Even the lowest power firearm loads have almost twice that velocity and generally about the same energy. A typical rifle cartridge (5.56x45, used in the M16 rifle and most NATO members' military rifles) has more than 8 times that velocity and more than 10 times the energy, and typical anti-materiel rifle cartridges (.50 BMG for example) travel almost 9 times as fast and carry more than 100 times the energy while still being portable enough to be considered 'light arms'.
Pretty much, unless you're doing some fancy materials science and creating armor that can only be penetrated using materials that can't tolerate the stress of being fired from a gun at high velocity, anything that can stop light arms fire is also going to stop arrows without much effort, and probably crossbow bolts too (which are much closer to the low-power firearm cartridges.
There are two other points to consider though regarding this final part of the question:
Other means of propelling a projectile reliably at low speed.
You've probably heard of BB guns, capguns, and airsoft guns. The first two are actually firearms (no matter what the US BATFE says). All three are capable of propelling small projectiles at speeds similar to an arrow, all of them are more reliable and easier to use than a bow, and all three are also more than capable of being lethal to humans if used right (you need damn good aim for that though, because you have to hit a critical part of the nervous system or the circulatory system and the part you hit can't be protected by bones). If you base your armor's abilities to deflect projectiles solely on speed, these will all get through it with no issue, and similarly basing properties on material that can only be damaged by some other material that's heat or velocity sensitive then these can probably get through it too.
On the note of railguns and coilguns.
There's a practical upper limit on how fast a railgun can get a projectile moving and still have it be at least slightly portable (resulting from a combination of issues with heat dissipation and energy consumption), and that limit using current technology is likely to be less than the speed and energy of a high-powered anti-materiel rifle (which would also be more portable). The few portable ones which have been made that used conventional projectiles also have fiendish recoil (as a matter of perspective, even a trained soldier can't safely handle the recoil from a .50 BMG rifle unless lying prone, and 20x102mm (the upper end for modern anti-materiel rifles) requires specialized recoil compensation to be safely usable even lying prone with the rifle braced properly, and any conventional railgun that can achieve the same results will have similar recoil (unless of course you decide to just eliminate Newton's third law, but if you do that you can also very easily make armor that behaves as you require).
Plasma railguns might work as an alternative (yes, those are a real thing) given that portable ones (usually small DIY projects) have rather low recoil, but they require even more energy than a conventional railgun (and if you can store that much energy portably, why not just convert your power cells to be used as grenades and be done with it?), and they have very low stopping power. A plasma railgun could be used as an incendiary weapon, but at that point it's still probably going to be inferior in terms of performance to high-explosive anti-tank rounds or more conventional incendiary rounds (especially when you consider white phosphorus and napalm as options), so you'd essentially need to handwave some special property of the armor that makes it weak to plasma weapons (perhaps something about the ionization properties)?
In short, short of some really complex physics we don't yet understand, this probably wouldn't work. Even accounting for that, it's not likely to result in the changes you suggest simply because things like guns are just too efficient compared to older technology.