your solution is pretty straightforward.
1. Buy a combined ADSL modem/router. Although you don't plan a wireless network now, you might find it useful in the future. Your children will grow, and have laptops, and want to use them in their rooms. Buy a BT voyager 2110 ADSL modem/router and you can have both wired and wireless access.
2. Plug the router into your BT phone socket (the one your present modem is using).
3. Connect it to your computer using the LAN cable supplied - you can plug the cable into any of the four LAN ports on the back of the router.
4. Go through the setup procedure as per the makers' instructions - it will be very simple.
5. Watch the router make the connection to the BT server.
6. Connect up to three more computers to the router using ordinary LAN cables plugged into the computers' network cards.
That's it - all the computers will be able to access the internet independently, without the need for another machine to be running. The router will automatically connect to the internet whenever you switch it on. Don't leave it permanently turned on, it's not good practice.