I had a 90-minute connection from an American Eagle flight to a BA-operated flight in T5. I used the bus from G17. It doesn't seem to be on a published schedule, but it took about 25 minutes for the bus to come (so, I must have just missed one). The bus didn't stop at the K gates and was empty when it arrived, so I suspect they use a different bus for the K and G gate routes. The total trip to T5 seemed to take about 5 minutes.
Between the Eagle flight being slightly late and the long wait for a bus, I showed up when the BA flight was nearly done boarding. (On the plus side, the bus drops you off near the BA gate in T5.)
I don't think I would do a 90-minute T3-T5 connection again, but at least the bus works as advertised. (And, I doubt I would have done any better using the train and clearing security again.)