I know BA on the 744 do T/O on flaps 20, climb flaps 10 to 4000' to meet the MNM alt. requiements and then continue acceleration and cleanup.
We do the above procedure for noise and environmental reasons, not SID gradient performance reasons (we climb at full CLB thrust until clean). In years gone by we used to use F20 to 1000'agl then clean up whilst accelerating, taking reduced CLB thrust with F5, and we still made the SID constraints.
The above applies to all weights and temps usually encountered at LHR in my experience.
I would anticipate the only problems likely would be following an engine failure, and then we would tell ATC if uable an altitude.