As promised, here is a quick link-fest of awesome Muslim initiatives:
The Safra Project. http://www.safraproject.org/ A UK organisation (mainly London-based) which supports LGBT Muslim women through counselling, creating safe social meetings, research projects to try and influence religious policy as well as providing resources for those wanting to learn more about lesbianism in Islam.
The Inclusive Mosque Initiative. http://inclusivemosqueinitiative.org/ London-based organisation which seeks to create a more inclusive Muslim community. They do a lot of educational seminars in London on Islamic issues and promote equality and diversity within the Muslim community.
The Muslim Youth Helpline. http://www.myh.org.uk/ This is primarily a telephone and online counselling service provided on a charitable basis for young British Muslims. I can admit to actually being a service user on a few occasions in the past when I was younger and always had positive experiences. It is like ChildLine, but service providers have more experience and training of cultural and religious implications/traditions/pressures on young British Muslims so the advice may be more suitable and service users are more likely to be understood better.
Imaan. http://www.imaan.org.uk/ Imaan are a welfare organisation for the British Muslim LGBTQI community. They provide counselling, safe social events, and an internet forum, as well as online resources for anyone looking to learn more about gender/sexuality in Islam.
Tell MAMA. http://tellmamauk.org/ Tell MAMA record hate crimes targeted at British Muslims and pass on this statistical information to interested parties such as the police force. They also provide support for victims of Islamaphobic attacks.
The Inter-Faith Network. http://www.interfaith.org.uk/ Obviously not just a Muslim group, and pretty self-explanatory.