I'd also expect that the Tiscali modem should work. To test it, create a new dial-up connection using the phone number 08456389800. Log-in with the username 'guest' and password 'webguest'.
That's the guest log-in for Webtribe. (That ISP has now been taken over by another firm, which in turn has been absorbed into yet another ISP, but the log-in still works from any phone number).
If the modem is working OK, you'll be connected to the internet at local call rates. You can then either install the Tesco software (safe in the knowledge that you've got a fully functioning modem) or simply allow your mother to continue logging into Webtribe (at the same rate as she'd be charged by Tesco). If your mother wanted email (without using Tesco's service) she could either use GMail or Gawab. (Both of those services offer POP3 mail, which can be accessed via Outlook Express, as well as web-based services). See here for further information regarding a Gawab account:
http://www.theanswerbank.co.uk/Technology/Inte rnet/Question407429.html
Chris