I made the rest of the IP changes, but a few hours later, I discovered that AOL and Comcast wouldn't accept email from me.
AOL amd Comcast (and probably others) check your reverse DNS (rDNS). So, if your server is called mail.something.com then you can check (for example, by pinging, or use dig or nslookup) that this resolves to the IP address 18.104.22.168 (I made this up).
Now, you're probably wondering what 22.214.171.124 resolves to. If you do
dig 126.96.36.199.in-addr.arpa ptr
Then dig will tell you. And if it doesn't give you a result, then AOL won't accept email from you. It's an attempt to reduce spam.
To make the rDNS work, you have to set up your DNS so that it works, and your ISP has to delegate the resolution of those numbers to you. I'd set it up, but my ISP hadn't done the delegation, they didn't realise that I was handling the rDNS. A phone call to Daisy soon sorted that out, and I now have reverse DNS. You can check it here:
So, now I've successfully changed all my IP addresses. My contract with Daisy expires in 18 months; if at that time fibre broadband is available to me, I'll probably switch to that, since I now know that I can handle the IP address changes involved. If I can do that, it will reduce my datacomms bill here from about £400/month, to maybe a tenth of that.