ETA if only exiting UK by air?

The ETA has nothing to do with leaving the UK - it's about arriving. It's linked to your passport and will be checked automatically when you go through UK passport control. Assuming you're taking the Eurostar, you're required to provide Advance Passenger Information in the same way you do with air travel. UK passport control is prior to departure.

Incidentally, noone should ask you for anything - it's all electronic.

All the info is on the web site:

https://www.eurostar.com/rw-en/travel-info/your-trip/travel-documents/advance-passenger-information

and yes you need the ETA before you check in at Eurostar

Doesn't matter the mode of transport (Plane, Train, Automobile, Boat, on Foot) the ETA needs to be attached to your passport. No one will ask for it, you really have nothing to show, but when they scan your passport, they know.

Will we be asked for our ETA as we depart LHR?

You won’t be at Heathrow if you don’t have an ETA when you board the train to the UK.