Nothing. Iceland already threatened to leave NATO over the issue, and the US bases in Iceland are critical to Atlantic defence.
If the UK doesn’t back down, Iceland threaten to leave NATO, and the US tells the UK to back down. But the UK knows that Atlantic defence trumps fishing rights anyway.
Iceland definitely isn’t falling in to Soviet sphere - they are pro neutral if not NATO. The US establishments and nato are useful financial income if nothing else, and if they aren’t bothered about the US sticking around it’ll be Nato in name only or fully neutral status.