AT&T cell phone

Here are the details of the Vodafone UK SIM card I have used on multiple trips with a Pay-as-you-go Plan:
https://community.ricksteves.com/travel-forum/tech-tips/vodafone-uk-sim-card-for-use-in-europe

The Vodafone UK global roaming page says that their plans work in Iceland with no additional roaming charges (beyond whatever plan you buy for the SIM):
https://www.vodafone.co.uk/help-and-information/costs-and-charges/travelling-abroad

I can’t speak to Iceland specifically, but I have used the ATT international plan several times in Europe and would highly recommend it. If you have an unlimited data plan, you can use your phone as if you were at home, WiFi or not. I’m curious to see what cell phone reception is like in general in Iceland, as we are going in August and I plan to use my ATT phone.

I used the AT&T international passport plan in Iceland in November 2019. Great coverage in the city (spotty in rural areas, but still not bad). The price is infuriating, but I travel solo and wanted to keep my own phone number on the road. Depending on how many days you're going to be there, the $70 passport plan (covers 30 days) is cheaper than the $10/day plan (if it even turns on for a minute, you're charged for the whole day). By the way, WiFi was everywhere including on both the small and big bus tours I took. In hindsight I might have done the $10/day plan and used WiFi for most things (but I was not driving so didn't need access to maps).

We have used our iPhones with AT&T service in Iceland several times with no issues. We do get the international plan and it worked well for us.