I have some archived flight data, and after questioning a controller who worked Gander Oceanic, he says that by time the flights were passed from New York ARTCC, they were all level at their final altitudes.
TAAMguy
The Oceanic guys issue the clearances to the flights well in advance of them reaching the oceanic boundary. The 'domestic' controllers and the pilots then between them, negotiate when any climb or descent will be carried out in order to present to the 'oceanic' controllers traffic in accordance with the clearance prior to the boundary.
In that sense, yes, the flights are at their final altitudes.
That, however, does not mean that they remain at that altitude for the duration of the crossing.
Do not forget either, that in NAT airspace normally on a given track the traffic is all in one direction, therefore separated vertically only by 1000ft rather than the rvsm 2000ft with bi-directional traffic, so the option exists for climbs of only 1000ft. In reality, it is unusual to have more than 1 step climb on a crossing.