How long will it take to repair a damaged iPad screen?

For practical purposes, in lieu of repairing and returning your own iPad, Apple usually exchange the iPad for a renewed/refurbished iPad of the same model and specification. The cost of the exchange is the same as would otherwise be charged for repair. Unless you have benefit of an active AppleCare plan, an out-of-warranty repair while expensive will be somewhat less than the cost of a replacement iPad.

As for out-of-warranty repair costs, Apple offers a fixed-price price repair. You can verify eligibility and cost for your region/country here:

iPad Repair & Service - Apple Support

If you have an active AppleCare+ plan, the cost of repair is reduced to a small excess/deductible fee.

The time required to process and exchange your device will be determined by availability of a replacement device. If an exchange device is available in-stock at the Apple Store or AASP, the exchange can be completed immediately. Otherwise, if a replacement must be ordered, it may take a few days to complete the exchange.

To add my U.S. repair experience to LotusPilot's excellent response:

In 2021 I sent an iPad 5 128GB for battery service and used Apple's mail-in/ship-in option. Apple directed me to a UPS store near my home. I dropped if off there on a Friday morning, and the store took care of packing it. Apple had already sent them the paperwork and shipping authorization.

Per emails from Apple, they received my old iPad on the following Monday morning and shipped the replacement the same day. They obviously had one available. The replacement arrived on my doorstep before noon on Wednesday. Six days—two of them weekend days— was pretty impressive.

Points:

—That was in the continental US. I do not know how it works in other countries.

—The replacement was perfect, both functionally and cosmetically. Apple installs a new battery and a new outer shelll (case+screen) in the refurbishment process.

—Apple will re-apply Apple engraving on personalized devices, but than can add up to 5 days to the process

and the BIG one:

—You MUST back up your iPad, preferably to a local computer, then erase it before returning it.

See: Sell, give away, or trade in your iPad - Apple Support