The question was since edited to ask for the "top" 3-4 mobile networks in the UK. That's a bit subjective, but answerable I think. The four below come first in most comparison tables (e.g. the Three one below), and in listings in resources like OpenSignal.com, and also matches my perception as a Brit of which are most prominent:
- O2 have been around for ages, and initially were set up from former national telecoms supplier BT. They've always been a big player, with prominent sponsorships etc.
- Vodafone, likewise, are long standing and also have a presence in many countries.
- EE are new but were formed from the merger of two other long-standing mobile companies, Orange and T-Mobile, and have a very extensive network
- Three are fairly well established, but newer than the others above. They've always pitched themselves as being a bit different - e.g. they were the first 3G-only network (now 3G and 4G only). Now they market themselves on being mobile data specialists and (more recently) for having the most simple and flexible pricing
Those are also the four that OpenSignal.com list, collapsing the others in an "Others" category, of which:
- Virgin are another long-standing big player but almost certainly the smallest of the big players, and I've heard more than a few complaints from friends who use them about signal problems
- Lycamobile and Lebara mobile are large international companies who appear to mostly market to immigrant communities on the basis of more competitive prices for international calls
- The others listed are relatively new, small or niche, or don't have networks of their own. That's not necessarily a reason to dismiss them, however, and they often get good signal through renting use of the bigger players' networks.
To get set up for smartphones, you'll want a 'Pay as you go' SIM card, and you'll want to check if the company does 'data bundles' (aka packages aka add-ons, various other marketing names). The other type, "Contract" SIMs in the UK, are generally 12 months minimum. Some companies do data only SIMs but these tend to be tied in to products like dongles and on a monthly rolling basis.
Most UK companies' Pay as you go SIM cards are free, you just pay for what you put on it.
You can buy SIM cards with usually £5-£20 credit already added in supermarkets, phone companies' own shops, and I believe WH Smiths at most London airports.
As for which company I'd recommend, I was very impressed with Three's pay as you go plan which I used for a few weeks in London:
Alternatively, for "mobile broadband" through a separate device like a dongle or mobile hotspot, add-on prices include: 7GB for £25 lasting 30 days or 3GB for £15 lasting 30 days.
Three have always positioned themselves as being all about 3g (hence the name) and (recently) 4g - consequently they had a bad reputation for coverage in rural areas in the UK, but in London, their 4g and 3g coverage is excellent.
You can check coverage in any area with the map at http://opensignal.com/coverage-maps/UK/ (thanks to CMaster for the link) - but everywhere in London gets good coverage (unless it's underground, or has thick concrete walls).