The wireless part of the internet has nothing to do with who your provider is - you need a router that provides the wireless part of the signal. The router itself plugs into the phone socket.
Depending on where you are you may have a choice between BT, what is basically the BT hardware in your local exchange but going to a different provider, or possibly there might be a provider that has its own equipment in your local exchange (that's the only way of getting a faster basic speed). You can find a lot of info about this stuff at
www.thinkbroadband.com/, including how to find information about your local exchange as well as various comparisons between providers.
What to look for:
- price
- reliability (outages, etc)
- service (call centre in India? UK-based helplines? Manned when?)
- speed (one could write a dissertation just about this)
- capacity (these days there's often a limit on how much bandwidth you can use without buying extra, if you download a lot of video or are into online gaming this might be an issue for you)
Any choice is some compromise between the above. Some of the cheaper providers have a very bad reputation regarding the other criteria. There have been a lot of take-overs, and quite a few of the original smaller providers are no longer independent. Plusnet has been bought by BT but seems to be kept separate so far and has a good reputation (but I'd worry about BT maintaining the level of service). My old ISP was taken over by TalkTalk at a time when they were expanding like crazy - I found I was suffering a lot of slow speeds for some things, but not others, which indicated to me that they were throttling certain kinds of traffic. I'm now with one of the smaller independent providers, but it's a bit more expensive. On the other hand the service has been excellent, and I've had very few outages (I need broadband in part for professional reasons so I can't really live with long outages). Since switching the speeds I experience are much faster (this provider promises they're not throttling traffic). I tend to think that BT is overpriced and the service isn't very good.
Usually you get discounts if you get broadband, phone and tv from the same provider, but beware that their packages might not suit you. (For example the package might look good but if you go above the agreed time limit phone calls might become very expensive.)
My recommendation would be to work out which of the above are most important to you and then look on the thinkbroadband site for recommendations.
Andrea