I’m not getting paid for this suggestion, but we found that the best option for us was giff gaff, because they will actually send you the SIM card ahead of time (for free), so you can set up your account before you get on the plane, and then once you arrive, replace your old SIM and get started with your new UK number. This will help with certain things, like setting up delivery appointments to your new place.
We like giff gaff because it’s cheap (£20 / month for unlimited texting and 9 GB of data, which throttles after that, but who uses that much data all the time?).
Just be sure your phone is unlocked. This might not work for you iPhone folks (alas, I’ve not been one of you for about seven years now), but if you’ve got an Android phone, it’s a great option.
Be sure to hold onto your old SIM for a bit. Once we arrived, we moved our old phone number to a Google voice account. It’s a one-time $20 fee, but it’s worth it for all those two-step verifications on US-based accounts (say, your bank) that won’t let you change to a non-US phone number. It also helps you keep your phone number for when you move back to the US. We’ll cover this more in the Upon Arrival section, but you’ll also want to hold onto your old SIM so that you can tether your computer to it when you do the Google voice set-up.