I've never talked to someone for it, and I would've first done it in something like 2008. (It's 'always' been done by SMS, well before 'apps' that might let you (and I don't recall it on websites).)
Well I haven't moved my number for a couple of decades at least, but the last time I did you had to phone them and go through the whole "are you sure? Let me see if I can get you a better deal" nonsense.
Cancellation is different to porting a number. You'll still need to cancel your previous plan, but legislation fixed carriers holding numbers hostage a while ago.
That seems like pretty low stakes – I wouldn't expect to talk to anyone to sign up and transfer to a consumer service in the UK. I don't think I've spoken to anyone for something like this in the last 10 years or so.