Generally speaking if a phone works with local carriers before installing /e/, it will also work with /e/. Here are two links for your reference: Carrier acceptance?
https://community.e.foundation/search?q=carrier
Further adding my own experience, I seem to remember that I started to use /e/ for my main phone when I was still with China Mobile HK. So, even anecdotally, at least CSL, SmartTone, and China Mobile are fine with /e/.
To sum-up my overall experience with /e/, I would say that, for my usage of a smartphone, with /e/ practically everything works as well or better than with Google Android. However, this will not be the case for everyone, at least at the current stage of development.
I mostly use open source alternatives to proprietary apps, and they all work as they should. So far, when I needed a non open source app, it worked without problems, except for e-banking apps. Many e-banking apps only accept the original Google Android as a security policy (notice, it’s mostly a policy…). As far as I understand, the /e/ team and various users are making a lot of progress in this area and perhaps, in a not distant future, most of such apps too will work.
For me this problem has been a blessing in disguise. I did not want to install e-banking apps on my phone, but I was under “family pressure”… As a solution, I took a good used phone (a Samsung J5), that had been retired by a family member, I clean it up of the installed garbage as much as possible (but leaving the original Google Android) and I made it my dedicated e-banking device. I use it without a SIM card (almost exclusively home WIFI), it does not have contacts, social apps, etc…
OK, the above is to give you an idea of how /e/ can be used with satisfaction and what not to expect.
Yuksing (Klaus)