A SID is a clearance.......comply with the clearance until you receive an instruction to do something else.
Question for you........if the controller had read out the clearance item by item including the step-climb limits, would you still pose the question?
OK, so let's just expand that:
[on ground]
"Fastair 123, after departure right turn to BPK climbing to altitude 4000 ft. After crossing BPK, climb altitude 5000 ft direct CLN." [FWIW, that's similar to a CLN6C out of Luton.]
[take-off]
"Fastair 123 contact London 118.825"
[frequency change]
Now consider three cases
1) "Fastair 123, good afternoon, London Control, climb to altitude 4000 ft"
2) "Fastair 123, good afternoon, London Control, continue the heading as a radar heading and climb to altitude 4000 ft"
3) "Fastair 123, good afternoon, London Control, climb to altitude 5000 ft"
For each case, what should my vertical profile be in the absence of further instructions?
Presumably we'd agree that in both cases 2 and 3, the original clearance is amended and no part of it still applies. What about in case 1 after BPK?