Originally Posted by aztimm

I just did the T3 to T5 connection yesterday, although from AY to BA, but I think still applies the same.

Fast Track cards are for immigration; they don't do anything for security. As a OWE, you can use Priority Security at T5, which may or may not be faster... You can bring a companion through with you. It puts you upstairs on the end of the security that those exiting the UK use. Usually that end area is just for Priority, but not necessarily.
I had 2:10 yesterday, and I needed nearly every minute of it. I was in biz on AY. We arrived into an end of T3, the walk to the bus seemed endless, I'd guess at least 15 min. Then the bus ride was another 10-15 min. Then plan for 10-15 min to get through security at T5, then a few more minutes to get to your gate. I arrived at the gate and boarded in 5-10 min, but I was on a 747, so they board those earlier.