After a month of luke-warm trickles for showers and washing up in cold water, we now have instant hot water from every (hot) tap in the house. It was a combination of problems.
1. The thermostat on the water heater had blown out, possibly due to the other problems.
2. The washer on the outlet of the water heater was too thick and had to be cut down to let a decent amount of water through.
3. I’m embarrassed to say that I caused the last and was so close to realising the cause, but never made the final step. It was clear from countless experiments that the hot and cold water feeds were mixing somewhere near the water heater. I assumed bad plumbing, but then why hadn’t the previous tenant ever complained? In fact it’s because we have 4 taps in each of our showers. One hot, one cold, one for the shower and one for the tap underneath. I was delighted to have a shower with preset temperature – just set the hot and cold taps and use the shower tap to adjust the pressure. Now you’ll immediately realise that this was where the water was mixing.
So obvious in hindsight.