East London Mosque will probably be the best for you inshaa'Allah.
As more people find out about Islam, some decide to convert. The East London Mosque provides advice, support and friendship. Weekly sessions are also provided so that they can learn to pray, read the Qur‘an and perform the other duties that Muslims are required to carry out.
Last time I heard (that was in Feb this year), East London Mosque were doing drop-in classes at 2pm every Saturday afternoon where anybody interested in Islam can simply turn up, learn about Islam, see the mosque, get literature, and afterwards ask questions on a one to one basis. etc. I'm not sure if they're still doing that, but here's their phone number: 020 7650 3000.
Even if they're not doing the classes, they'll still be most welcoming to you. In that case, phone them and arrange an appointment. Explain that you would like to get some more information on Islam, and may consider reverting and is it possible for you to come and speak to someone that can spare some time to help. Then one of the imams may either arrange a time themselves, or may arrange for one of the knowledgable sisters to speak to you, or they may even say just pop in at any time. I think they will probably have literature that you can take away and read too. As a note, there is a chance the phone may not be answered during prayer times (times are displayed on the website), just try again later if it isn't.
The contact details, if you need them, are here: http://www.eastlondonmosque.org.uk/contact