They should be able to do it together. Also - if anyone brings in an iPhone with a cracked screen for a battery replacement, I've been told that they simply won't do it without the customer agreeing to pay for a screen assembly.
It might take a tiny bit longer since they would need to disconnect the touchscreen connector and reconnect it to the replacement part. Also - a new touchscreen has a new, preinstalled home button, and they would definitely need to place it on their rig (the "Horizon Machine"). I heard that any kind of service where the screen is replaced gets this treatment, but some claim that a battery service alone doesn't get it. At the very least they're going to place a new adhesive seal.
However, there are no guarantees about how long this would take. The work itself to disassemble, remove, install, and reassemble might be about 15-30 minutes. But then there's the placing on the rig to test it out plus get the home button talking to the CPU/security ICs. I think that takes about 10-15 minutes. And who knows how long depending on whether or not they have the parts in stock as well as available technicians and work backlog. It can be done and ready same day, but a lot of people with late appointments find that it's not ready for pickup until the next day.
There are other possibilities, including receiving a complete service replacement for the cost of the requested repair service(s). That's the fastest way that they "service" a device and still the only way that iPads are "repaired" by Apple. They'll send it to a place to get it repaired/refurbished where it will likely become someone else's service replacement. Definitely make sure to back it up before bringing it in. Even when they do an actual service and return the same device, the terms of repair service say that they could wipe it for various reasons.
