I've realized that I should be using Electrum instead (but, despite some great tutorials, I couldn't install it)
It really doesn't require a tutorial to install and set up. The only issues I can think of that prevented you is if you are using an outdated desktop OS, like Windows 7 and older, or an unsupported and old Android version. Electrum is sadly not available for iOS.
And finally, I've got some NEOs in an Atomic Wallet and I didn't dare look to see if they were still there...What should I do?
Firstly, ensure you still have access to them and that they are still in your wallet where you left them. I see no point in being afraid to do it. If they are there, great. If not, there is nothing you can do about it.
And above all, what can I do? Swap neo for btc? Send them to a neo wallet? Or something else?
That all depends on how much you have and if there is any interest in buying them. You can always sell them on a CEX or perhaps a DEX if there is a market for them. P2P or instant exchanges are also a possibility. Seeing that you need gas to move your NEO, go and get that first.
Can I use the private keys in a neo wallet for example?
Does Atomic Wallet give you access to private keys of individual coins or do you only have the seed? You can import the seed elsewhere, but then there is the question of derivation paths and what Atomic Wallet uses vs what your other client uses. Again, if you have no gas to pay for the network fees, importing your seed into another wallet is not going to help.