Doctor Who has always had a large LGBTQ fan following, even before the show returned in 2005 under Russell T. Davies. When Davies took over as showrunner, fresh off Queer as Folk, which made him the hottest showrunner in the UK who could write his own ticket, he chose to bring back Doctor Who. It was a huge gamble because no one in the industry was sure the show would succeed even though the British audience's hunger for the show never went away.
Davies made a point of introducing LGBTQ characters to the show to reflect the social realities of a diverse Britain. He also cast a lot more actors of colour in the show to that end. He introduced the hugely popular Captain Jack Harkness (John Barrowman), the first openly LGBTQ character to the show, and had many LGBTQ supporting and guest characters to the story throughout his run. Homophobes accused the creators of pushing a "gay agenda" on the show. Yet when Steven Moffat took over as showrunner and continued to feature LGBTQ characters on the show, nobody complained. Madame Vastra (Neve McIntosh) and Jenny Flint (Catrin Stewart) were very popular amongst fans.
The biggest potential change was when Chris Chibnall cast Jodie Whittaker in 2018 and declared at New York Comic Con during her debut as the new Doctor was that "the Doctor is nonbinary". That was a direct quote, yet the show has done nothing with the Doctor's status ever since. No details in the show ever addressed that, nor, strangely, have any outlets, even LGBTQ sites or magazines talked about it or did interviews about it. Whittaker's Doctor has been the most queer-coded Doctor of them all. She eschews a dress or any fashion that suggests traditional femininity apart from her haircut. All her T-shirts had a rainbow on it, and that's not an accident.
Whittaker's Doctor feels like we barely know her because there were never any stories or key moments that truly revealed her core the way Christopher Eccleston, David Tennant, Matt Smith, even John Hurt, and Peter Capaldi's Doctors did. Under Chibnall, the show did its best to take everything except the Doctor's story. Yaz's (Mandip Gil) unrequited romantic feelings for the Doctor come too little too late when Whittaker only has 2 specials left before she regenerates and leaves the show. It's not Whittaker's fault – she's a perfectly good actor who was more than willing to explore all aspects of the Doctor including her relationship with Yaz. They still skirt around it when Attitude magazine interviews them about it, and it's the only time they've actually talked about the LGBTQ aspects of the show. The show had previously been happy to play up Tennant as a heterosexual male romantic lead for female fans and continued with Smith and Capaldi but has ignored that aspect of Whittaker's Doctor. Hardcore fans prefer to think of the Doctor in any version as asexual, but even that is an LGBTQA+ category now. Chibnall opened that door, but it seems he and the BBC have strenuously stayed away from it ever since. It's their way of having their cake and eating it. Everyone else on the show can be LGBTQ except the Doctor.