I just had a similar issue with the hot water in the bathroom sink.
Shower was OK. I messed around for a few hours trying to backflush the
galvanized pipe, replaced the 4" nipple that collapsed from rust, etc.
I ended up having to replace the pipe back to the water heater with
copper. This was a 1950s vintage house and you could not see light
through a 4" section of pipe it was so corroded.

It is probably the galvanized. If you have already checked the faucet
and other easy things, it is probably time to replace some pipe,
especially at 80 years old.