Why would the SIM be different? I've had the same SIM since I joined TM, and it's been through Motorola, BB, Samsung, Nokia. Also the Nokia and one Samsung were unlocked, and European. In fact, the Galaxy 3 was in Czech, which changed to English as soon as I put the SIM in. The Motorola RIZR was the first phone.
BB has it's own system, for push and a TM BB usually has UMA (wifi). - If your SIM came with the BB, and you had BB BES, BIS, I suppose it could be programmed in.
My unlocked phones used EDGE, and got internet just fine.
I could get online with my own router using the BB UMA and my settings, but damned if I could do it anywhere else. (Except a TM hotspot)
There is supposedly a different plan for Android and BB, but this shouldn't affect wifi.
If you had a BB data plan, then you might get a notice about switching. Did you have a data plan with the BB?
It is possible to go without a plan on an unlocked phone. You have to enable wifi.
Z