Verizon has a CDMA network and AT&T has a GSM. They are incompatible. Even if you have say a T-Mobile GSM phone, it's not going to work on AT&T's GSM network, nor is it going to work on Vodaphone's GSM network in the UK, because in the US the carriers hard code the phones to their networks, and the FCC has regulated different "bands" of GSM here than what is in use in Europe and the rest of the world.
I do think Verizon and Sprint, since they both use the same CDMA data frequencies, do share data networks in many places, but the voice CDMA2000 part of your phone is still hard locked to your respective network.
If you get an unlocked GSM phone in the UK, you CAN use that phone with other carriers in the UK, because they don't make them locked down to specific networks there.