What Happened to Heathrow Passengers without an ETA During the Blackout

if they have a valid passport, then there wouldn't be any problem

The ETA is not a condition of entry; certain visitors are meant to have an ETA prior to travelling to the UK - that's not the same thing

Almost certainly no-one was affected. Heathrow has a night flying curfew due to how many residents live under the approach paths. The fire service were called at 2323 just as the curfew came into force, and the loss of power would have been later than that.
Thus no-one would have had a connecting flight that late at night as the last flights had already departed. Although the airport doesn't close as such you can't stay airside overnight.
So anyone connecting overnight would have been landside (with a valid ETA) by then in ay event.

They would simply talk to an actual border agent.

The ETA is part of the larger automation of the entry process, with an ETA, you use the e-gates your info is matched to your ETA and biometrics, and on your way.

Short of any requirement, a person is involved

And of course, not all visitors need an ETA - some already have visas, some have Right of Abode, etc.

Was there enough power for border force to process arriving passengers? Or did they have to revert to some back up total manual non electronic method?

Was there enough power for border force to process arriving passengers? Or did they have to revert to some back up total manual non electronic method?

As stated above this is a totally theoretical question. Power was lost sometime after midnight. Due to the night flying curfew the last flight of the night had arrived at 2255, so everyone had been processed through by that time- in fact it is very likely everyone had been processed at the time the fire started.
The last flight had also departed, so border control and security was closed at the time that power was lost.

The first inbound next day arrives at around 5am.

There are no flight movements between those times unless (as happened the following night) special authorisation is given. Even when that happens it is for a very limited number of movements only.

or to answer the headline question in a very few words - nothing - it didn't happen - there were no passengers