I vividly recall being on a T5 to T3 bus. There was a luggage truck in front of us, and the people next to me on the bus said to each other "isn't that our bag on that wagon?". So yes, bags do move faster through LHR than humans. There is no right answer here, your connections are entirely do-able and normally bags make it. You could leave a vastly longer connection and still have the same problem. But connections are always the weak spot with checked bags, so you best arrange yourself to be able to cope without checked bags for a day or two at your final destination, whatever the scenario.