Just to be really clear, if you go through LHR - transit or stopover - there are two main, high taxes that can be applicable:
- UK Air Passenger Duty (coded as GB): chargeable depending on distance band, measured between London and the capital city of the final country of the journey, as well as class of service (Economy or higher-than-Economy). Note that as of 1 April 2015, the number of bands will be reduced to two, meaning that the maximum APD that can be levied from thereon will be GBP 138 (notwithstanding any indexation or increases). APD not applicable on international to international transits (less than 24 hours) through a UK airport on a single ticket.
- UK Passenger Service Charge (coded as UB): chargeable on all flights which go through a UK airport - transit or stopover. Not dependent on class of service. A quick ITA search shows the charge is GBP 42.06.
Any sectors where you fly BA (or IB) you will have the fuel surcharges (coded YQ) passed onto you.
As I said in some posts a while back, even if these charges are high, don't get too coy or uptight in trying to avoid them. For example, if you want (or need) to go to London, but decide to land somewhere else in Europe as your stopover in order to avoid the APD and PSC, then consider the cost (and time) of getting from your stopover point to London (and back to resume your itinerary). If the cost of the (re)positioning transport (train or air) is not cheaper than the APD + PSC, then there's really no point avoiding London except if you're doing it on principle.
Again with BA - as I've said before - there's little good in saying "there's no options to go" or "I won't go" simply because your only options are on BA and you have to stump up extra cash.