Unfortunately as mentioned most internet in the UK is ADSL which requires you to have a house phone. I know a few people that went with BT initially before switching to TalkTalk, but for the £99 connection and the rate you'd pay for TalkTalk per month (just under £20 for home phone & internet - your line rental is inclusive in the price) + any monthly usage. However you'll be locked into a contract with BT (most likely a year to 18 months) before being able to switch to TalkTalk or you'll automatically be in a contract with TalkTalk. We used to have BT with Plusnet, then switched to BT for home phone & internet (cost about £50 a month!), then eventually switched completely to TalkTalk which is about £19 / month.
I know PlusNet formerly a Sheffield company are now owned by BT used to have pretty reasonable rates for home phone & internet, but personally I prefer TalkTalk for their inclusive International Package compared to paying an extra £5 on top of everything else and internet separate price package from BT...
If you haven't already checked you may want to look at
Broadband Checker to see what's available in your area and see if anyone has any home phone & internet packages. I know O2 and Orange had some pretty good packages (with discounts for their contract mobile customers), but didn't really look further into them as TalkTalk worked out cheapest for us personally. If you can get a Sky package for home phone, internet & TV that may be good, however your £12 line rental is on top of their package price, not inclusive plus if you want the International Talk option its an added £5) so price adds up, but could still be cheaper than having a separate home phone / internet bundle on top of your Sky subscription.
As Shahbanou mentioned Mobile Broadband is an option, however in addition to the issue of connection you'd be more than likely paying a higher rate for it than getting a home phone & internet package.
One of the pros to having a landline:
Believe me if you ever have to call any companies for their Customer Service you'll want a home phone to use instead of a mobile since pretty much no companies have 0800 (toll-free, except from using a mobile - you still have to pay) lines anymore they're all 0870 / 0845 etc. which charge to use and from a mobile you're looking at extortionate rates compared to using a landline. This is even applicable to some chain stores instead of having direct numbers.
Good luck!