I would be looking for other causes, rather than the paint. Is there moisture in that area, where are the two closets in relation to each other, and any number of things.

Looks like a really bad paint job, with zero sanding or priming, and the layers are delaminating due to moisture.

Could this at all be caused by underlying lead paint? The house was built in 1952. I know lead paint can have alligatoring appearance sometimes but not sure if thats relevant here.

Likely caused because the paint layer underneath wasn't allowed to thoroughly dry before it was top coated. Or possibly the wrong type of thinner in the paint base coat. Lucky that it is hidden away in a closet.

What would you recommend the solution to be to have new coat look good? What do I need to do before applying fresh coat of paint? From the few patches where the grey paint has flaked off, the underlying plaster walls have the same hairline cracks in them. Thank you!

Well, it's a closet. Once you store stuff in it, you won't be able to see much of the wall surface...even if you looked at it. Which you won't because you'll be looking for something in the closet.

But, since we now know that the paint is telegraphing cracks in the plaster, you need to rip the plaster off and drywall it for it to 'look good'. Very likely the baseboard will need to be replaced, too. Paint isn't the 'look good' fix.

FWIW, the finish of the plaster in the closet probably wasn't as nicely done as the rest of the rooms when the house was brand spanking new.