@CheeseMmmm
Reason T was included is that gay men have long history of cross dressing eg drag. Part of history etc and if man caught cross dressing was evidence of gay and therefore dangerous arrested etc.
I believe in the context of the UK this is correct, but as we know it today the T in the LGBT movement, and the LGBT movement in general, mostly comes from activism and social organization in the United States.
I'm not sure about the exact makeup of the participants of the stonewall Riots but if I remember my history correctly it was actually a mixed crowd between gay men, lesbians, and "trans people" then conceptualized sometimes as transexuals or just plain crossdressers. Basically, the Stonewall Inn (located in the Greenwich Village neighborhood of Manhattan NYC), was generally a bar for those who did not fit the social roles of society. The cops tried to break it up, everyone fought back, and as the story goes the LGBT movement (with everyone participating) was formed. This also coincided with a general American bent towards liberation and activism sparked by the civil rights movement. So, at the same Time you had the Occupation of Alcatraz by AIM (the American Indian Movement), and a lot of activism around women's rights all coalescing around the same time.
The T in LGBT has always been there, but what the T stands for and how that section of the community has always been somewhat shifting, and only recently has been codified, specifically, as "Transgender," but they've always been members of the community, and self identified as women, but were not seen so by larger society, and had little means to organize with each other to create their own groups.
This can be seen in the documentary film Paris is Burning. It follows the drag/Ball culture of New York City in the 1990s. Notably, though not much distinction is made between Drag Queens and Trans Women, both exist as separate entities both participating, with Venus Xtravaganza being the main trans woman followed (she would, by the end of the documentary, be found murdered under the bed of a motel).
Trans Women were also a large part of the AIDS movement/crisis in the 80s. Many trans women were particularly vulnerable as quite a few resorted to prostitution as no one else would hire them, and therefore were (like gay men) vulnerable to AIDS and died in very very very high numbers.
So, basically, in the American movements Trans people have always played a part in these social movements.
As I understand it, in the UK there was less specific community based organization around LGBT issues, and the acronym is somewhat of an import from the United States. Much of the focus was on Buggery laws for gay men (though I don't know much about how it effected Lesbians in the UK). But, broadly, the UK didn't go through those same social reforms and wasn't subject to the same movements as the United States, so the culture around LGBT is different, but also influenced by, American LGBT movements, causing a bit of a disconnect.