See this link about moving your service from Virgin to BT
You will be given a new router called a Homehub which is configured to work out of the box so no need to set it up. It is easy enough to connect PC and Smartphones.
You will have access to BTWifi as BT Openzone is now known.
I am unable to help regarding migrating your web hosting.
If you don't want to use the homehub, you don't have to, any third party ADSL router will do fine. This has the advantage that you can use BT wifi whilst out and about but don't actually transmit it yourself. For web hosting, 1&1 are ok.
The transfer normally goes well, as we found when we transfered from Virgin to BT. Virgin have to issue the MAC code and when you've passed that on to BT they will do what is needed to transfer your service. It took about three weeks for us. The speed you will get depends on many factors, such as whether your local exchange is running 21CN equipment (so can use ADSL2+), length of lines from exchange, etc. Note that the 'sync speed' often quoted is not the actual data transfer speed, that is always slower (ours syncs at 3392 kbps but transfer is currently 2.7 Mbps but varies slightly). Also note that it can take at least 10 days for your speed to 'settle' and sometimes longer.
I cannot comment on web hosting as we do not use any such service and see no need of a personal 'home page'.
BT will send you a new modem/router, the Home Hub, but you pay a small carriage fee to receive it. You don't have to use it, but the service, in our experience, is faster and more reliable with the HH in use (something slightly odd about the way BT Internet is set to operate I suspect). You can turn off the broadcasting of the FON WiFi service as well as the internal WiFi provision - without affecting your ability to connect to other FON services available when you are away from home. I live in a very rural hamlet and there is absolutely no point in wasting electricity leaving the FON transmissions running, so that is turned off (easily done from the admin menus). Likewise, we do not use WiFi from the HH as its location very close to our Master Socket means the WiFi signals do not reach the rest of the bungalow, so we use Ethernet (recommended for when you run speed tests at http://diagnostics.bt.com/login/?workflow=Speed) to connect normally but do have an access point running 802.11n at 300 Mbps for visitors and mobile/tablet usage.
Also worth noting that you can set preferred IP Addresses for equipment so it always has the same network address. I have a networked print server (with two different printers connected) and that works best with a 'fixed' IP, so that is set after you have set up the system (easily done again by looking at the Home Network under Settings (you have to loggin and go to Advanced), clicking on each item shows the allocated IP which you can change and/or set as the default for that item.
Enjoy your broadband but don't expect miracle speeds.
Thank you, Mike: that's very useful indeed. Great to hear from someone who's made the same move!