How does a hot air system heat water for showers etc?
It doesn't, that bit is simple, I had a hot air system in the first house we had, it seems likely it was damaged on the installation, it was too easy for someone to step on the ducts, so we found it expensive to run, but that may have been because damaged.
Back then,1977 I think, as in that house in the Winter of Discontent which I think was 1978, the windows were single glazed, and there was no other heating in the house. Some good points, and some bad points, from turn on to warm house it was fast, much faster than water systems, but also cooled down fast.
Today this may be good. However, back then, the vents in the walls would not turn fully off, so all rooms were heated even if you tried to turn it off. All the doors had vents, to allow the air to return to the furnace, can't really call it a boiler, as no water to boil, and it made the house very dry, we would put trays of water in the air intake.
With the electric being turned off, during the Winter of Discontent, we got very cold, as even if gas fired, with no electric it will not run, so we moved house, to one with a flue gas fire, which had not electric required, so we could keep warm in a power cut, which after Maggie crushed the Unions, did not really happen again anyway.
As to domestic hot water, the immersion heater is not as expensive, as to what I first thought. A boiler, be it gas or oil (Yes I know they don't boil water any more, but we still call them boilers) has a large heat exchanger which needs heating up, so there are a lot of losses when using a boiler. An immersion heater will also have loses, but it depends on the size of the tank, and how much insulation.
We have two extremes with DHW (domestic hot water) washing hands, and having a bath. One is using maybe a litre of water, the other maybe 40 gallons, and the old idea was only heating the top of the tank, unless having a bath.

The Irish worked out how to heat varying amounts with the Willis system, but we were too tick to work that out in the rest of the UK. The other method is to heat the water local to the taps, fitting these
under the sink, resulted in no wait for hot water, and reduced loses, but no good to fill a bath, I have two instant electric showers, also two baths in the house, but the baths are never used, however I have 5 sinks, so fitting 5 units starts to look rather expensive, so I use a single central hot water cylinder

and in summer it is heated with electric, from the PV solar panels, so the unit which does that, also records how much used, so this shows

how much used in 7 days before yesterday, at 25p/kWh looking at around £2.30 per week, for me with PV solar and off-peak, looking at around £1 a week in lost revenue as not exporting the solar. So 5 x £100 for units at each sink, does not make sense.
Returning to the hot air heating, while we were both at work, it worked well, as it heated the house so fast, we simply turned it on when we got home, but once we had our first child, so heating wanted 24/7, then it worked out expensive.