Basic summary of what's going on:

• Allison Bailey is a lesbian woman who helped set up LGB Alliance (though she has stated that she technically wasn't a founder).

• A couple years ago now, Allison made two tweets about some of her pro-LGB views. Specifically, she tweeted about the disgustingly homophobic "Cotton Ceiling" concept-- the idea that "trans lesbians" should "break through the cotton ceiling of" the underwear of lesbian women, who don't want to date them.

• Allison got harassed by trans activists and had a campaign directed at her in an attempt to get damage her workplace reputation and get her removed from her workplace. (She's technically self employed I think, but is a part of a group of lawyers/barristers/whatever legal people they have in England, called Garden Court Chambers, which can kick her out.)

• Allison decided to sue her workplace and also Stonewall, because Stonewall had become closely associated with her workplace and some of the people at Stonewall-- who also had associations with other LGBTQ groups, apparently-- were trying to get her kicked out of her lawyer's association.

That's the gist of the situation, as I understand it (as a non-lawyer, non-UK curious person). I've been following the trial by reading the transcripts of it that get posted-- check out Tribunal Tweets if you're curious.

edit: Whoops, meant to address your actual question. LGB Alliance is a group that works for lesbian/gay/bisexual (same sex attracted) people's rights. They criticize the pressure put on children and LGB people to transition.

I was originally wary of LGB Alliance, I'll admit, given that so many other organizations claim to support LGB people but actually don't. Having kept track of them over the last couple of years, though, I think they've done a good job overall of focusing on LGB rights and well-being and not bowing to any particular ideology.

are they finally having the trial about the charitable status?

It looks like they are and that this got sorted out almost 1 year ago? Source: https://www.gov.uk/government/publications/lgb-alliance