To share some recent experiences (as someone who has to notify).

Eurostar - quick look at my passport both ends only. No questions, no 'discrete' checks. Same in both Paris and Brussels.

Eurotunnel with car - quick look on way out, nothing more. Lot longer process on way back - passport obviously flagged on UKBA system and the guy didn't really know what to do, so a long time asking all the usual questions - where had we been, how long away etc. and he wrote all the answers down by hand on paper. Not a big issue as my fellow travellers aware of the situation