This is a tricky one - First, check out the list of available handsets: http://www.voipon.co.uk/wireless-ip-pho ... arison.php
You obviously need a WiFi capable handset as a DECT handset (as you pointed out in your OP) will not meet your needs. This handset will also need to be registered with a local VoIP PBX such as asterisk or VoIP provider such as Voipfone using your local WiFi network. Next:
You now need to look at connecting your legacy PBX equipment to the same VoIP PBX as your WiFi phone. I don't know anything about the BT Versatility PBX your using. From what I read online it is possible to connect it to asterisk if you have the Broadband plus module? You can configure up to IP 12 trunks from one or more Asterisks. However, you can't connect 3rd party IP terminals back to the Versatility with BT firmware loaded.
What is the actual model number of your PBX? Can you find out what firmware version it's running?
Once your Versatility PBX is setup and communicating with your Asterisk box or chosen VoIP provider, set all inbound calls coming in on trunk 1 to the extension of your choice on your Versatility PBX and the same for trunk 2. Now simply set a speed dial on your new WiFi phone to call either trunk 1 or 2 and the relevant extension connected to your Versatility PBX will ring.