You seek the holy grail of phone service; a number that works worldwide. Based on many years of trying various services for this purpose, here are the best options I have found.
Skype (More Convenient, Less Reliable)
First signed up for a Skype number years ago for inbound calls and once had as many as four numbers simultaneously in different countries.
When paired with Skype credit, for outbound calls and texts, this was a massive improvement compared to anything else on the market previously. Unlimited outbound calls are available as well as fairly inexpensive per use options, so it is easy to find an option that meets just about anyone's specific voice calling needs regardless of location or country of origin, etc.
The advantage of Skype numbers are that they are easy and straightforward. You often don't need citizenship, residency, or even to go to a country. Some countries have more arduous requirements, but for many, fill out a form; boom, you're done.
The disadvantage of Skype numbers is that they always have been closer to a landline in capability. You can't port in an existing mobile number (only a landline), although it looks like you can port out a Skype number these days.
Originally, it was possible to send SMS, but not receive SMS at all. Then, at some point, it became possible to receive some SMS messages, but it wasn't particularly reliable (and naturally, you never know how many messages you didn't receive; some people would ask, though, and nope... didn't get that one). It looks like Skype formally supports inbound SMS on US numbers now, so let's hope they show up, but "identity verification" messages officially are not supported.
Based on real-world experience, I don't think a 2FA code via SMS ever arrived on a US Skype number — although voice call verification generally worked if that was an option — but I stopped trying to use it for that purpose years ago. As an experiment, I just gave a US Skype number a shot with a 2FA code for an account where I know it didn't used to work and it did arrive safely, though. Perhaps the odds have improved.
If you want to use Skype, would suggest that you sign up for a US number as an experiment and see if inbound SMS works with specific accounts that you use, but don't expect it to be 100% reliable.
UK & US Cellular Carrier Services (Less Convenient, More Reliable)
Unfortunately, the real world is comprised of many different systems,
some of which are connected, some sort-of connected, some not
connected. While it's possible that the complete interconnectedness
you want might be achieved in the future, it isn't here now.
As @DavidSupportsMonica wisely points out (above), this is true. Based on personal experience, rather than try to go for complete interconnectedness, the best option is to bypass cellular entirely with a service that operates anywhere with Wi-Fi (which is just about as close as you can get to global connectivity).
Have tried services that receive calls/texts in some places and not others via various degrees of roaming (sort-of connected), but if you travel frequently, this is almost worse than not being able to receive calls/texts on a number. In practice, it means that you can go from being well connected to falling off the face of the earth for a week or month; never good. You run the risk of losing your number, as well.
You're wise to be leery of Google Fi, they apparently will discontinue your service if you have been outside of the US for some unspecified period of time.
Because you mention the United Kingdom, specifically, the current best option for your needs or anyone else who wants a UK number (don't believe UK citizenship/residency is required), likely is Xpatfone (which runs on top of Devyce, a similar UK service with options that are a bit different). Devyce promises support for the EU and Brazil soon, too.
Have not used this service personally (and have no affiliation with it), but a British friend based in Spain uses it for just about everything you mention without issue. You can port in a UK mobile number and receive 2FA via SMS. It runs on top of local cellular data or Wi-Fi.
For those with a US residence, the best option I have found is Tello. Actually tried a couple of other sort-of connected options and went back to Tello. Tello is inexpensive, geared toward international use, routes calls/SMS over Wi-Fi when you are outside of the US, and has worked with every 2FA ever thrown at it. The only practical annoyance in real-world use is that if you haven't connected to a US network for three months, it clears a prepaid balance to zero and sends a scary email about disconnecting your number (but it actually tops up US$10 automatically without issue if you have set it to do so); the minimum cost effectively is US$40/year. Monthly plans for higher usage also are available.
I believe this is acceptable promotion because it also benefits the reader, but for those with a US residence who would like to try Tello, you can start with a US$10 credit with this link (and in the interest of full disclosure, I would receive the same).
Another inexpensive option for those with a US residence to consider is UltraMobile PayGo (no affiliation). It does support international roaming for a lot of destinations (220, reportedly) and likely would be best for someone going outside of the US to locations where support is guaranteed to be offered (rather than extensively traveling around). Curiously, it does not support domestic roaming inside the US. Actually bought a SIM card for this service but have yet to use it because Tello added the Wi-Fi connectivity option and it was sufficient.
Happy, well-connected travels!