What's a gay transsexual woman?

A transsexual woman is:

a male-to-female (MTF) transsexual or transgender person and the term trans woman is preferred by some individuals over various medical terms.

Because a transsexual woman would view herself as of the female gender (though of the male sex), the term "gay" applies as it would to any other woman. That is, a gay transsexual woman is a gay woman, who happened to be born as a man.

As for your original questions --

Is it a man who became a woman, then turned gay and started liking other women? Or is it a woman who likes other women (thus is gay), and then turned into a man?

You cannot, simply by the English language, tell which came first. Sexuality and gender identity are fluid and personal, and standard phrases (such as "gay transsexual woman") will not be able to tell you whether the individual first realized that they were gay, or first realized that they gender-identified as a woman. Only the individual, if they feel comfortable telling you this, will be able to explain.

What if you changed the word order, and got a "transsexual gay woman"? Then breaking it down, you have a gay woman. This woman is also a transsexual. But you don't know which part came first still--were they a transsexual woman who later realized she was gay? Maybe. Perhaps they just feel more comfortable identifying as a gay woman, who is transsexual. This is their identity, but the term does not capture which came first.

In my comment, I said that people don't turn gay, and upon further reflection, I can see that in this case, you could make that argument.

Gender and sexuality are separate from each other. Gender is whether we identify ourselves as male or female. Sexuality is whether we are sexually attracted to women or men.

A gay trans-sexual woman is a genetic male who identifies as female but is sexually attracted to women. Before she identified as female, she (as a man) would have still been attracted to women, and therefore would have been heterosexual. Changing her gender identity "turned her gay" but it didn't actually change anything except how she identifies herself.

Had she been a gay man who changed her gender identity, she would now be considered a heterosexual trans-sexual woman. In other words, she would be a genetic male who identifies as female and is attracted to men.

So yes, the term is unambiguous.

Generally, a transsexual woman is someone who comports herself as a woman, and was considered by others to be male earlier in life. (That is, less ambiguously, a male-to-female transsexual.) The idea is that, to the person, she is a woman, and the transsexual part is merely a descriptor.

How you then apply "gay" to that I'm not totally sure. I suppose if you were to consider "transsexual" to be merely a descriptor, then it could be dropped (at the expense of being less precise), and then you'd get "gay woman", which is unambiguously a woman with sexual interest in other women.

A bit of updating is needed here, because terminology is changing over time. At this point, transsexual is starting to lose currency in favor of transgender. I think the shift does improve clarity somewhat.

A transgender woman (now commonly called a transwoman) is a genetic male who identifies herself as and knows herself to be female. And by the way, this holds true whether she has begun any degree of physical change towards any version of typical socially recognizable female appearance or not. If she has made the physical change, it is easier for those around her to understand, of course, but it is no more or less valid either way.

If she is gay, it means she is sexually attracted to females (of any kind).

So a gay transwoman (what you were originally calling a gay transsexual woman in this question), is what you might say is a lesbian who began life as a genetic male.

A final note: In One Person, John Irving's latest novel, is an insightful and sympathetic exploration of almost every person's life that could possibly have been touched by this and any other LGBTQ issue in the last half a century. In one way or another, that amounts to just about all of us. I urge everyone to read it.