Who knows; a second; a day; a month maybe.
You certainly need to change the drive as quickly as possible, but in the meantime, make sure you do have ALL your personal data backed up.
Physically changing it is no problem. A few screws, a jumper and a couple of connectors and that's it.
Just how your manufacturer arranged the recovery option, remain to be seen. Some have a hidden partition that contains essential data, others don't.
I doubt that the drive is "keyed" to the recovery disc, as this would be too restrictive for repairs such as yours.
I suspect, when you change the drive and put your recovery disc in the computer, it will reload the operating system. But it will be as when you received the machine new. Any extra software, will have to be reinstalled.
Some drive manufacturers offer free utilities to clone the contents of your existing drive to the new one. This is OK, but you get "warts and all", so any stray files and so on, that inevitably collect over time, will be transferred as well. It may also lead to start-up problems, as your new drive will be different to that which XP expects. If that happens, XP's "Repair" option may sort it out.