The short answer is that it depends. (Ain't that always the short answer!)
If you feel that this information will be relevant to your potential future employment at the company, you should try to find a way to disclose during the interview process.
By "relevant", I'm essentially talking about (and please excuse my crassness) the concept of "passing". If you feel like people are going to wonder about this type of thing, and it's not just a piece of information you can disclose to coworkers you trust in private, then I think it will be easier for you down the line if you find a way to disclose early on.
If you don't feel like that will be an issue, I see no reason to tell anyone anything for any reason other than that you trust them and want them to know something cool about you.
It might not be an issue if you're interviewing, say, at an LGBT+ advocacy outfit where you can reasonably expect that folks will know what is and isn't rude to point out or ask, and that they'll be understanding if they inadvertently offend you. But if you're interviewing at an old and prestigious financial institution (for example) you might unfortunately encounter some ignorance (or downright hostility). That's the case in which I feel it might be important to disclose ahead of time. By doing so you'll give the company a chance to brush up on its legal obligations as far as gender discrimination go and maybe even warn your future colleagues not to be jerks.
Sidebar: There are other issues in play here, of course. In the U.S., it's frequently considered inadvisable to disclose any non-obvious demographic about yourself, since employers are strictly forbidden from discussing those things with you in your interview process. (They can't ask you about your religion, marital status, age, etc - lest they decide not to hire you after learning that info and leave themselves open to a possible discrimination suit.) I don't know about the U.K. but I can't imagine it's too wildly different.
Here's what I would recommend: Disclose your situation to your recruiter after you have passed the initial phone screen and are moving on to interviews. (If you're not working with a recruiter, this might be trickier.) Here's why I think that would work best:
- Disclosing trans-ness to a recruiter means you're disclosing to someone with some HR training. They are most likely to know the laws and the rules, and to be sensitive to your situation.
- Disclosing after the initial phone screen means that you will be disclosing after the recruiter has made a "hire" decision about you, so you're saving them the awkward situation of having it look like they made a decision based on the information you disclosed. Additionally, they'll now be in a position to prepare your subsequent interviewers not to react poorly or make uncomfortable comments.
- The recruiter will be performing your reference check, and will therefore be in a position to handle that more effectively. That will avoid the "Toni? We never had a Toni working here..." situation that makes it look like you lied on your CV.
I wouldn't say that this is the ultimate solution, though. If you're on the phone with the recruiter for the initial screening and they sound confused at hearing your voice, it might be worth dropping in a mention that you used to go by
_______. You don't have to disclose explicitly, but you're acknowledging their confusion and confirming that they're not making it up. Then after you've aced the phone screen and can be a little more candid with the recruiter you can tell them whatever you feel they need to know.
Disclose early, but not immediately, and if possible disclose to an HR professional first.
Disclaimer for anyone reading this outside of the U.K.: check the discrimination laws in your area before disclosing. In many places in the U.S. it's still completely legal to fire (or not hire) someone for being trans. Obviously the situation is vastly more complicated in a situation like that.