If the OP is not able to convince the check-in agents at SYD to check the bag(s) all the way to LHR, the OP needs to enter Taiwan (pass through immigration) to pick-up the bag(s) from the bagage belts, proceed through customs to landside and find the check-in counters of the TPE-LHR flight. I doubt transfer desk staff will be prepared to pick-up the bag(s) and retag them to LHR.

With 2h25min at TPE it should be possible but there is not much room for delay of incoming flight.

There is no way to link or combine separate tickets/reservations. If you miss the TPE-LHR flight you need to rebook it according to ticket rules (it will be considered voluntary rebooking)