I've coincidentally always had female T's and male pdocs (although I was my first 2 pdocs only a few times each. And my current pdoc's sessions with me are rather therapy-y). Funnily enough, I can't imagine it any other way.
Well - I can't imagine having a male T, so I don't know whether it would work. And I can't imagine having any other pdoc (first because I'm scared of doctors and not (anymore) of him; second because our treatment, with us doing therapy-y things, is pretty much unique around here - pdocs are mostly med management, part/leader of the treatment team, and may do therapy-y things in a crisis, but not for 3-4 years the way we do; third because I really like him (as in, how he acts in session). He's extremely normalizing and has only overreacted once. And he's VERY anti-force, big on autonomy and everything).

But anyway, around here choices are limited, especially for pdocs, so you'll mostly have to deal with whoever you are assigned. Basically, less complicated cases are dealt with by small practices (and only then if the GP or his staff can't handle them), and complicated cases by big practices. You can choose which practice to go to - and in small practices you might also be able to switch T's more easily (of the 2 or 3 T's present), or switch to a different practice altogether (as they generally have a short, or no, waiting list: big ones are supposed to start your treatment within 14 weeks, but half a year is much more likely). More of a 'you're the consumer, you're paying, you decide'. In big T organizations/clinics, you are basically assigned someone and have to deal with them. It's not easy to switch T's if the T feels you two still have opportunities - if they think a different T (new insight, more expertise, different modality) might help, they would probably want to set you up with a different T. You have some input of course but it's rather limited. Also, waiting lists are LONG and most clinics only have 1 pdoc anyway per location (and if they have more, there'd usually be 1 for depression, 1 for autism, 1 for .. So you'll go to the one they decide fits your problems the best) anyway. More of a 'insurance pays us to do a job and we decide how we do that' thing.

By the way, my (our) family T is male, I've had some trauma therapy from a male T (I did have more input in choosing my trauma T's btw, was allowed to 'vet' them for the most part), and when I was inpatient this summer my doctor (I think you English lot call it a psychiatrics resident? Someone who is a basic doctor, and studying to become a pdoc. Is supervised by a pdoc) was female.

Eta: health insurance (there's basic and extra, basic is mandatory by law) covers a lot around here, there are limitations of course but generally you could say that if you have a DSM diagnosis, you should be able to get the evidence-based treatments covered 100% by contracted practicioners and depending on your policy, partly to fully by non-contracted ones. (Partly - I think 75% from up to the rate the company lists as normal - in cheap policies, fully ('free doctor's choice') in more expensive policies (also up to the normal rate)) No referral = no coverage (and GP's are not supposed to refer you if it's not necessary, e.g. mild or recent depression or anxiety/panic would first be dealt with by them and their office staff), no DSM diagnosis = no coverage, no evidence-based treatment = no coverage although there's an exception for if there is no evidence-based treatment (then the treatment research most indicates that works will be covered), and most meds can be prescribed off-label and will be covered like on-label (which can be anything from no coverage to copay to full coverage). The ministry decides on the basic health insurance minimum coverages, although they are allowed to cover more (not less). And there's a deductible.
Result is that waiting lists are LONG especially for psych care, and that a lot of providers (basically all that need a referral to go to, which is mostly everyone but physical therapists - although that isn't really covered by basic insurance (there are a few exceptions for partial coverage by certain conditions) anyway) see it as your privilege that they allow you to come, not as their privilege that you pay them. So you have to take what you're offered and be satisfied with that. (You can voice your preferences, but they might or might not be listened to. Most likely 'not'.)