I have already solved this, but I thought I would post it in case anyone else has the same problem.
I have multiple email accounts, with gmail, yahoo and Demon UK. I have my gmail app (bundled with the phone) set to scan for new gmail emails every few minutes, and my email app (bundled with the phone) set to manually download Demon emails, yahoo emails and also a second copy of any gmail emails.
After the ICS update, the gmail app continued to work normally but the email app would not load new emails, and when the "sync" button is selected (which is meant to load new messages) nothing happens. It doesn't even go looking for messages.
I solved this by changing the settings (one at a time for every email account) to download new emails automatically (I set it to "every hour"). I then manually downloaded new emails, which is still permitted when auto-download is selected, and all my new emails arrived as they should.
After that I selected it all back to manual, because I don't want to use up my mobile data allocation (I occasionally get a lot of emails).
So now I am back where I started, except it all works.