[MovedGoalPosts]

Right, now I'm clear on what you are intending.

1) Ignore, scrap, sell, whatever the internal PCI ADSL modem. You willl not need it again.

2) For your existing PC, make sure it has a standard ethernet network card. If it doesn't simple ones cost around £10.00

3) Buy yourself a wireless router.
This wireless router can be with or without an integral ADSL modem. If it has a modem integral you will not need your BT modem. If you do not have an integral modem you will need the BT modem. I do not know if the BT modem uses an ethernet or USB connection. If it is USB it will probably not be compatible with a wireless router so in which case you should get a combined wireless router with ADSL modem.

4) Buy a wireless network adaptor card for your new PC.

5) The wiring setup will now be:
Wall socket of incoming ADSL connection > modem cable > ADSL modem
ADSL modem > ethernet cable > wireless router
(if you have a combined wireless router modem then it will be wall socket > modem cable > router)
wireless router > ethernet cable > original PC network card
wireless router > no wires needed > new PC wireless network adaptor.

This setup has the advantage both PCs can now operate independently of each other, but you can transfer data (slowly) between them. Both PCs can be switched on and off independently of each other.

Since the original PC is located in the study, where you are already happy to have a wired setup there is no point in giving it the added expense of wireless. There is also nothing to be gained by switching back to your internal ADSL modem, when you give up AOL. Your BT modem should work quite adequately on any ADSL service, unless it is limited in some way on it's maximum speed service. You 'd just need to reprogram it, when the time comes, for the new ADSL supplier's logon.