How the ****...?

I have when that happens!

We had a dead bird stuck in the chimney once, so I can relate somewhat...

I can see how it could get in there, because indoor hot water heaters have to have a floo to let the gases escape, and the cover might have come off and allowed the bird entry. However, I'm fairly certain that a dead, boiling bird would concoct some much richer aromas than musty, and that after a month of it boiling near constantly, you would not be able to tell what bird it is or that it's a bird at all without a degree in anthropology or ornithology

At least in the USA this would be absolutely impossible to happen. The only water connections to a hot water heater are two small water pipes. No bird could ever get inside the water tank under any circumstances. Gas water heaters have a flue (exhaust pipe) but it’s only about 3” in diameter and it only connects to the burning chamber - there is no connection to the water which is separated from the burning chamber. Sea gulls are large birds and could not even fit in the flu. I suspect this is from some place where they collect rainwater in a cistern or storage tank. That could be open and you can get critters in a cistern. But the cistern is not a water heater and people do not generally use cistern water for consumption for just these reasons. It would typically be used for flushing toilets and possibly washing clothes. OP - I think you misunderstood what the plumber told you or you misstated the problem.