I know how bad it is when your e-mail is not reliable (which is one reason I have mail go to Runbox and Google Mail, in case one is down, and important 'support' mail goes also to my freebie PAYG Orange mail address, so I get a text alert).
Having had my first e-mail address for 8 years on domain ultranet.com which 'went' during a takeover by cable firm RCN from Chicago, I decided an ISP e-mail address would no longer be a good idea.
In the short term, you may want to use www.mail2web.com but in the longer term, I'd get a free account at Google Mail, or Yahoo! mail, or www.Live.co.uk if you use Windows Live. I use a few commercial e-mail services but also like to use Google Mail as a reserve as it is free!
I know that after so many years you don't want to change it, but you are no longer using Tesco.Net (and most of the ISPs could demand a fee for using their mail service, and bump it up every year).
ISPs often seem to rely on people staying loyal because of the 'hassle' of switching.
You've made the switch of ISP, and have a {temporary} problem, but what if Tesco.Net decided to close their mail service ?
In my case the ISP was taken over, in some other cases, such as CloudNine, they closed down. In the case of Tesco.Net, they could, too. No joke, they recently withdrew their Voice over IP service, which they used to promote on TV, because the phone numbers were allocated to a company which has ceased trading, so Tesco cannot run it, and all those phone numbers and unusable, and the VoIP phones probably difficult to change to use anything else.