I think, in the best case (the right people are all together, and the right raw materials are readily available), you could probably get a modern computer built in 10 years, and a relatively basic microprocessor built in a few years less. If we don't have to reinvent a global communications system to get experts talking to each other, build mines and refineries to produce chip-grade silicon or rebuild airplanes to get materials from around the world, this task is a lot simpler.

The biggest problem isn't that modern computers depend on some piece of technology or process that is lost to us. It's that all the people who understand every step in that chain, from survivalist blacksmiths who can smelt ore to make tools to industrial chemists who can fashion substances that nature doesn't readily provide to electrical engineers and developers who will actually program the damn computer in year 9, are currently widely dispersed. Put them together and give them what they need and I think they could get the job done more quickly than you think.

Of course, it's more likely that everything and everyone you need will be scattered around the world when this instant occurs, and if that's the case, I'd honestly bet it would take hundreds of years to independently reinvent a modern computer because you'd need to reinvent most of modern society to gather the people and the materials to do that.