ON a trip we did recently in F which involved LHR, PHL, MCO, PHL, LHR we eventually took to checking in the buggy as all it did was delay us getting off the plane, negating the benefits of F priority disembarking. Worst was LHR to PHL where waiting for the buggy meant we then had over an hour in the immigration queue.
On the last leg PHL to LHR it was checked through, it took 15 minutes from the door opening to getting in the taxi.
Sling until they're too big and then just carry them is my suggestion. Check/ditch the buggy :-)