The reason you would have been given the instruction "climb in accordance with the SID" is that your initial instruction on the ground from the Tower would have been something like "After departure maintain altitude 3000 feet on reaching".
This is because you were probably released by Thames onto their frequency with a 3000 foot restriction.
Just after you got airborne, the Thames controller obviously decided that he did not need to work you after all, and that you could be transfered straight from the Tower frequency to London Control. At this point the Thames controller would also have asked the Tower controller removed the 3000 foot restriction from the release, hence the phrase "climb in accordance with the SID".
I am sure that if you then received any heading instruction from London Control, it should most certainly have been accompanied by a level instruction, as the SID altitudes only apply to the SID itself.