There are no "pubs" open until the early hours. Only recently in England have the licensing laws changed which allows them to stay open until midnight. A lot of pubs still have the traditional 11pm closing time. Because Central London is such a residential area, only clubs with soundproofing or away from residential buildings will be allowed a late license.
For clubs it depends what kind of music you like and what night of the week you like to go out. Roadhouse in Covent Garden is open until 2:30am Mon - Wed and 3am Thurs - Sat. You are unlikely to find anywhere open after 1:30am on a Sunday night. Tiger Tiger, On Anon and Zoo are open all nights of the week.
A lot of clubs in London stay open until 3am on Friday and Saturday (in fact almost all so impossible to list).
www.viewlondon.com
www.londonnet.co.uk
for club listings
The dance music clubs stay open pretty much all night at the weekend
fabric
http://www.viewlondon.co.uk/clubs/fabric… stays open until 8am on Sat night and 5am other nights.
ministry of Sound, Pacha etc. although you might not be served alcohol after 2 or 3am