Is there anything I could inadvertently do that would cause me to lose the ability to claim BCH (and maybe other forks)? There's isn't anything like a claiming window or deadline, is there? I'm mainly interested in claiming BCH.
No, you can't lose the ability to claim BCH. After the hard fork, all addresses that contained any amounts of BTC received an identical amount of BCH. That BCH will reside at the address until someone claims them.
Just keep your private keys and seed phrases safe so that no one has access to them.
I'm keen to move my BTC to my Nano, and that's my first priority, but I don't want to rush, in case the exact sequence of doing things in order to claim fork coins is important.
You don't have to rush, there is no time limit.