Make sure to check if you have any other existing APN profiles! If you do, you have to remove them.
To check: Settings > Mobile Service > Mobile Data Network. Make sure the APN under Mobile Data and MMS are both three.co.uk. If they are not, then you might have another APN profile still added.
How I removed it (for iPhone): Settings > General > VPN & Device Management > Profile you want to remove > Remove profile.
Solved the issues I had with mobile data issues straight away. Might save someone else from going through an endless cycle of rebooting and resetting!
(More context below, from my experience)
On holiday with the family and my dad is the main account older for our SIMs. We were in Italy for a holiday, so we had Go Roam activated across them all. However, upon arrival, two out of the four devices were functioning properly, while the other two had 3G blinking on and off in the corner. I could get the 3G to stay permanently but the internet just didn't work (on full bars too). We ended up spending a couple hours talking with staff trying to figure out a solution - checking mobile data was turned on (it was), data roaming was turned on (it was), turning airplane mode on and on, rebooting the phones several times, resetting network settings, changing to an unlimited spending cap, all to no avail.
We took a break from Three support, but I spent a while thinking that there was something the two phones had in common, with which I was suddenly reminded that me and my dad had both used a SIM card in Japan, which required us to create a new APN profile for it to work. I was searching left and right how to remove an APN profile but Google wasn't giving me the result I needed. However, it also turned out I wasn't looking in the right place - I kept looking at the options in Mobile Service. After a quick search with the network provider I had in Japan, it was as simple as mentioned above. As soon as I removed the profile, my mobile data immediately started working.
I'm not familiar with the technical terms, but that profile seemed to be overriding/hindering Three's APN profile, which made the mobile data under Three not work.
Quick side note, I felt like there were times the support team felt a bit exhausted by this (as were we though), but at times it felt like they thought we were simply just doing something wrong, hence the endless cycle of checks and resets. I'm just glad I managed to figure it out eventually!
I am far from being a professional with this sort of stuff, so if any issues are still unsolved, I'm afraid I will probably not be able to help much further. Hope this can help!
