It's usually possible to change to a pay as you go dialup on your old ISP, and then pick up POP3 mail using your new broadband connection, or even have a premium mail account that allows you to send as well as receive POP3 mail through your old ISP. Using the first option you need to dial up (yes, dial up) to the old ISP about once every 6 weeks to keep the payg account active. The second is usually about £1.50 per month - oh wow!

Then you put a note of the new email address in your signature, and gradually everyone shifts over.