I don't know if moving from the U.K. is anything like moving from the U.S., but I would advise leaving plenty of time - the paperwork from UTL takes a good while. It's never too soon to get started. Good luck!
if you're not fussy about being out in the middle of (relatively) nowhere you might consider some of the smaller towns, like Borgarnes, Egilsstair, ═safj÷rur, Bl÷nduˇs, Akranes. It'll be harder to find work in these places though, and I have no idea about schools there.
You are correct that you don't need to do the massive paperwork stack when coming from the UK, as a fellow EEA member. The rules for you are here: