This will be partially familiar to those of us who did BA domestics out of T1.

The process is, and my estimated timings as someone who is, dare I say it, fit:

T1 domestic arrival gate
Up the T1 domestic terminal towards the exit - 2 mins
Turn left and up an escalator to a higher section - 1 min
Long walk (or slow moving walkway) to Flight Connections - 4 mins
Boarding pass check at Flight Connections - 1 min
Down escalators from Flight Connections to the bus area - 2 mins
Wait for bus at the T5 specified gate - could be 10 mins (15 min service frequency is advertised, but it seems to be sub 12 mins in reality).

Bus transfer to T5 - 12 mins

Into T5 and up escalator to Flight Connections area - 5 mins
Over to the far North end for T5 transfer arrival - 2 mins
[Look like you know what you are doing, the BAA staff may home in on your and pen you in for a bit if it is very busy in T5]

Assuming you are NOT onwards travelling in domestic (i.e. you have an international departure from T5):

Conformance - 1 minute - you need to be here [in general terms] 35 minutes before your flight or you will be offloaded.
Boarding pass check - 2 minutes
Pre security check point - 2 minutes [But can be longer if they are trying to regulate the queue up the escalator]
Up escalator - 2 minutes
North security - 5 minutes [but can vary from 1 to 10 minutes]
T5 Gate - between 5 minutes (domestic A gates) and 20 minutes (end of C satellite - note there are further ups and downs if using the transit shuttle to gates B and C).

In all cases there are lifts as alternatives to escalators, they won't save any time with the exception of those going to the B and C transits.

I have made a flight from T1 to T5 in well under an hour, but I know exactly what I am doing. But put like this it seems a lot of flapping about, I don't really notice it. Land side takes longer and there's plenty of up and down with that too.