I've never seen a LL or KK departure still at the SID levels overhead Dover. They are usually above FL240 by this point and being transferred to Maastricht for higher.
The worst affected SIDS are SS departures via Dover as they get stuck underneath the LL arrivals from the east and then are usually climbed underneath LL and KK departures routeing the same way. So are frequently still at FL80 passing Detling. However, this is all set to change when LAMP comes in supposedly late next year (I think).