iMac Late 2013 21.5" does not boot after re-assembly, fan just blows

The 21.5 iMacs were not user serviceable.

Macs are extremely sensitive to having the correct RAM modules and just shoving in generic PC RAM will burn you. If you swapped out the internal drive and put in a new one without the power sensing cable, the fan will just blast as the Mac has no means to detect its drive temperature.

If nothing works after putting back the original RAM I would guess some assembly error in one of the cables/connector pins Even if the RAM was an issue the screen should still come on with error message if all else is correct.

As tbirdvet already said, I also think it might be some RAM misconfiguration. Although very unlikely, it could also be because the config was reset when you took out the coin battery. Just retry the old RAM, maybe it could be a broken socket or something

Tried booting macOS Recovery?

How to start up from macOS Recovery - Apple Support

Follow the instructions for Intel-based Mac (obviously).

If I had to guess, any number of incredibly tiny connectors may have been dislodged or is broken. Working on these Macs requires microsurgical skill.

As an aside do not use generic PC RAM on Macs. Macs are intolerant of RAM that does not actually meet its requirements. PCs can cope with poor quality RAM, Macs can't. To my knowledge there is only one source for Mac memory that actually meets Apple's stringent requirements, so if you didn't obtain it from them, it probably won't work. Be sure to revert any questionable configuration to its original operational state. The reason for pointing that out is that your troubles with iMac #2 immediately followed upgrading its memory. Other than those sockets, the logic board shouldn't have been affected at all. Removing the screen for internal cleaning and NVRAM battery replacement may have broken something though. Disassemble, inspect, reassemble, hope for the best.

Try resetting NVRAM: Reset NVRAM on your Mac - Apple Support. Contrary to popular belief there is no need to do that twice, three times, n times, whatever. Your observations of Step 3 will confirm it was performed correctly.

SAkiwirunner wrote:

That fan blowing all the time must mean something,

It means that one of the system sensors is not responding. Macs will run the fans at high speed in order to protect the system if one of the sensors is out of range or not responding.

but I am accepting defeat and will likely trash these Mac's - too much wasted time already. I've fixed all kinds of electronic equipment, built computers over the years, and dismantled and fixed Android phones with teeny tiny connectors. For the Macs to stop working just because they were disassembled is crazy - the connectors are small but straightforward

These iMacs are very delicate. There are multiple very fragile & delicate cables & connectors that are very easily damaged even when you are being very careful. It is also difficult to fully seat some of those cables, or you may have forgotten to reconnect one of them.

If you removed the Logic Board, then you could have accidentally damaged it as well since it is a very tight fit when removing & reinstalling those boards.

FYI, there are also 4 diagnostic LEDs located on the Logic Board of the iMacs which can help to identify power issues and GPU & LCD issues.