The SmartPhone Networks In The UK

In the UK, GSM tends to be the standard for the smartphone networks, and these networks offer text, data & minute allowances per month for pay monthly contracts, and for some pay as you go, or sim only deals, although some allowances can be monitored daily for PAYG.

The main networks are as follows:

O2 (previously BT Cellnet & Cellnet) - This network is owned by the Telefonica
EE (previously T Mobile & Orange) - This network is owned by the BT Group
3 / Three (previously 3G & Hutchison 3g) This network is owned by the CK Hutchison Group
Vodafone - Vodafone

There are then virtual networks that literally "buy space"  from the above networks.

The virtual smartphone networks include:

1p Mobile - Uses the EE network
360Coms Telecom  - Uses the Vodafone network  - Upfront payment for 3 / 6  or 12 months
AfriMobile - Uses Three Network 
Age UK My Phone - Uses The Vodafone network
Airwave Smart Mobile - Uses the EE network
AlwaysOnline Wireless - Uses the Three Network   
Anywhere Sim - Uses O2, Vodafone, EE, Three networks 
Asda Mobile - Uses the EE network - now defunct
Axis Telecom - Uses the EE network
BT  - Uses the EE network & offers sim only or pay monthly phone deals
C4C Mobile - Uses The O2 network  
Call & Give  - Uses the EE network 
Champions Mobile - Uses The O2 network
CTExcel -  - Uses the EE network  
Delight Mobile -  Uses the EE network  
Econet Mobile  - Uses the EE network
Ecomy Mobile  - Uses the EE network   
Ecotalk - Uses Three Network
ETS Mobile - Uses the EE network
FreedomPop - Uses The Three Network  
Fome Mobile - Uses The O2 network  
Gamma Telecom - Uses The Three Network
Giff Gaff  - Uses The O2 network 
HP Mobile Connect - Uses the EE network  
iD Mobile - Uses Three Network
JOi Telecom - Uses Three Network     
Jump O2, Vodafone, EE, Three    
KC Mobile - Uses The O2 network
Lebara Mobile Uses the Vodafone Network  
Lycamobile - Uses The O2 network
Meem Mobile - Uses the EE network     
Natterbox - Uses the EE network
rdTelekom - Uses the EE network
w PAYG - Uses the EE network
Pebble Mobile Network - Uses O2, Vodafone, EE, Three Networks
Plusnet Mobile - Uses the EE network & offer 1 month sim only contracts
Rok Mobile - Uses Three Network
RWG Mobile - Uses Three Network
Sky - Uses The O2 network
Smarty - Uses the EE network
Talk Home Mobile - Uses the EE network
Talkmobile -Uses the Vodafone network (now owned by Vodafone & Pay monthly only)
TalkTalk Mobile - Vodafone transitioning to O2 network 
Telecom Plus - Uses the EE network
Telfoni - Uses Three Network
Tello - Uses Three Network
Tesco Mobile - Uses The O2 network
The People's Operator - Uses Three Network
The Phone Co-op - Uses the EE network  Vodafone (Vodafone with fairphone handset only)    
Toggle Mobile - Uses The O2 network
Truphone - Uses The O2 network
U2i Mobile - Uses the EE network
Vectone Mobile - Uses the EE network
Virgin Media (Previously Virgin Mobile) - Uses the EE network
Vivio - Uses the Vodafone network
VoIP.co.uk (Firstcom Europe) - Uses Three Network
White Mobile  - Uses the EE network
These virtual networks range in size, customer base & the products that they specialise in. For example, some offer cheap UK bundles, whereas others focus upon European deals.  Some provide 2g and others are on 3g or 4g. Some are on all 3.

Some of the networks focus on a sim only monthly payment plan, ranging from 1 month to 12 months, whereas others focus on a flat rate pay as you go amount, with variations between the cost of the calls, sms & data bundles.

Others perform credit searches, some have a prepayment upfront system so there is no credit score.
Depending upon your own needs & requirements, certain network deals will be more suitable for some people, so it pays to do your research.