Unfortunately we will never get to the level of speed that we are now, and things will be incredibly complicated to do. Now, once any known programming language and (by some miracle) a monitor is implemented to work with a mechanical computer that runs exactly one program at a time on it written in that language we are essentially at a level relative to (I believe) that mid-60's early 70's in terms of computer science. The reason for this is that computer science does not care about hardware. It only cares about what basic operations the hardware performs, which make up its machine code.
The core problem here is one that plagued early computers that were mechanical rather than digital, which is that you cannot have binary. Now don't get me wrong, you could compute the binary representation of some number or vice versa. The issue is that memory was encoded via gears. Have you ever seen a 2 sided gear? It's a bad idea. This means that things like adding circuits and other basic operations have to be done for several cases. However, when we do it in binary we can think of everything as a series of component wise boolean operations on vectors of true/false values. This is essentially the great benefit of electrical computing and it's great hurdle. When people thought to make the switch they were forced to do binary because of there were only two wavelengths that the hardware could produce. That led to a somewhat better way of thinking about such things. Ironically most routers use 32 or sometimes 64 different waves to compress the data down since it is now feasible to do that sort of thing.
Now turing machines, automata, algorithms, programming languages and a lot of the other more design oriented concepts might advance the same. They are still concepts interesting to study and they lie more on the end of computational theory rather than actually implementing concepts or testing them on a machine.
Unfortunately regardless of mechanical speed or durability you have a fatal flaw in your mechanical machines which mean that computers will never develop to where they are now without a switch to electrical or some augmentation.
Light consists of photons which can be produced/released by a release of energy. Light is how computers communicate (wi-fi, cellular signals, radio waves, etc.). It also how we are able to see the monitor. The fundamental flaw is that machines without electricity will not be able to produce light waves like this. It just doesn't happen. Otherwise ordinary fires and humans walking around could create interference. You need electricity to do that. Therefore your mechanical computer world will not develop computer science to the strength it is today. The reason why that will happen is because there will be no monitors. There will be no personal computers. There will be no internet. Taking this out of the picture will reduce your world's level of advancement in every field drastically. Plus, without people taking an interest in computer science that are not in government positions or in large research facilities... you've heavily limited the number of people who might advance the field by a large number. It might even be considered nothing more than what it was meant for, which was a tool for performing mathematical computation. Nothing more, and nothing less.
There will also be no sophisticated flying machines to speak of. Without the advanced systems there are now, passenger jets would be off the table. This also heavily drops the level of advancement in your world.
There would be no word processors, no hand calculators to speak of, and no databases. Even if the latter two existed, the former would be limited to complex series of operations being entered into a typewriter to be read back through and print a document.
Ultimately, anything can be made with your mechanical computers that anyone else can make with an electrical computer assuming the former is turing-complete. However, the issue isn't one of capability. It's an issue of it being unfeasible and uninterpretable. Imagine computing a number and a series of pegs raising into the air to represent the binary version of the number. It would be a very strange calculator. Without monitors, editing documents of any form would be challenging.
Interesting enough though, the only issue for blind users would be the lack of a sound interface. However, I could see such people adapting quite well to a mechanical interface that uses raised and lowered portions to form a "monitor".
In fact, I will correct myself from earlier. We can have monochrome screens, but that is it. Essentially we can raise and lower portions of a flat plane to make an image.
Ultimately though without the invention of the internet, this site wouldn't exist. And therefore there it one thing that your mechanical computers will never have.