We fell foul of this last year arriving into T3 with a tandem. Not sure I can answer all your questions because the web pages I got information from seem to have disappeared, but from what I can remember...
This is a temporary arrangement while the tunnels are refurbished. The intention is to bring the service tunnels back into use as a cycle route when the refurbishment is finished. Not sure if its going to be one dedicated tunnel two way for cyclists (which I encountered in 2014) or one way tunnels shared with taxis as before.
For us, the best route in/out was via T5 and Heathrow Express (fast, frequent and free between terminals). Cycle paths to T5 are excellent.
Heathrow seem genuinely keen that their workforce cycle (to do with targets they have for air pollution/CO2 I think). I think the bus you mentioned is primarily for Heathrow workers who are encouraged to park at the Cycle Hub and get the bus in. (Not sure if it's the same bus to actually take your bike in.) It might be worth contacting the Cycle Hub with your questions. http://www.heathrow.com/company/heathrow-jobs/commuting-to-heathrow/cycle-hub
This page http://www.airportwatch.org.uk/2015/08/about-85-million-spent-to-update-heathrow-tunnels-and-reinforce-them-against-heavy-a380s-above/
says the work is to be completed this month, so depending on when you are travelling, you may not have a problem.