Using US cell phone in London: Verizon's Intl data cost can be a rip-off. Here is our experience in case it is helpful. Our visit was 30 days in September, 2018. Using iPhone 6.
After much research, I bought a SIM card (EE) in London for 15 GBP for 5 GB of data. It was easy to install and worked with zero problems. We made about 20 local calls (out of 500 in the plan). Had good data service in London, Cambridge, Portsmouth and Windsor.
A key issue is texting. Texts to our US number do not come through because the phone was on the GB number. Fortunately for us, all of our family have iPhones, and iMessage texts do come through.
Another result: we only used about 0.3 GB of data out of the 5 GB allowed. We used Google Maps, Safari etc as much as we wanted, and still only used 0.3 GB. (Of course we were on WIFI in the apartment.) I believe we saved money versus Verizon's plan but because of low data usage, not a much savings as I thought. My guess is that for a shorter stay, it wouldn't be worth the trouble to get the English SIM. Of course I am comparing with Verizon's INTL plan which has little data allowance.
Final thought: upon leaving I wondered if EE would try to automatically renew our plan. We had paid with cash, so they had no card to charge. I was told in the store, it would not automatically renew. Switching back to the US SIM was super simple.
Hope this is helpful for some travelers.