So I reinstalled it again because they still appeared offline when they weren't, and now I can select the realms :)
They aren't saying they're offline anymore, I can switch realms fine ^^

So basically, I turned the game off and on again a couple of times - solves all computer problems xD
I hope ...

So for the past 2 days, it's just kept going to the realm selection screen, so I've been unable to play. :(
I've tried everything I can think of:
The realmlist.wtf is definitely not read only.
The launcher is fully up to date.
I also tried to re-install the game.

Edit: After re-installing it, it ...

When I look at the realm selection screen, it says they're both offline - so I can't select them in order to connect to them, even though they're actually online. Normally this isn't an issue - usually when I log in, it goes straight to character selection and I can play no worries - it still says ...