Sadly, it is hugely expensive to access the internet without either a phone line (line rental of at least �10.50 per month) or Cable.
Cable will provide a broadband service without the telephone or tv, but it is likely to more.
Talk talk may be an option - it works out around �17 - �20 per month including line rental, for 8 meg broadband. All land line phone calls are included, so you may save on your mobile phone bill.
All others (as far as I know) quote prices WITHOUT the line rental.
If you do find a very cheap option, I'd be interested to know about it. :)