It's a Baxi 105e, so not the most elderly or the most recent out there,
but I have no reason to suspect there's anything wrong with it.

Turn on a hot tap, and it heats some water. Quite a lot of water, in
fact, as it has an internal hot tank which ought to supply you with a
basin full of water without having to fire up.

Except that it always fires up, and the internal tank is cold by the
next time you want to wash your hands. So it short cycles, and heats
water you don't need.

Then there's the shower. Combis are really good for showers aren't they?

Not this one. Maybe it will be better in the winter, but at this time of
year with warm inflow the minimum burn rate is more power than the
shower on full. So after a while it gets too hot, shuts off the burner,
and runs on the internal tank. It does this until the internal tank has
cooled a bit, then fires up again to heat some more water.

Mean time the shower water temperature is going up and down by 10C or so

And the other morning when the water pressure was a bit low for some
reason? Wouldn't fire at all, unless the shower was on full hot.
Whereupon it wound the water up to 60 or so, then turned off again.

I'm glad this is a short term rent, and I won't be here long. On the
other hand, the place we are moving to has oil...

Andy