I haven't applied for mine yet because I'm still married & divorce is a bit too expensive for me at the moment (and I'll be darned if I'm going to ask his permission to legally change my gender). But I'll presume you're fully aware that you can't get a new birth certificate until you've obtained a Gender Recognition Certificate.
However, I don't know where you've heard of £3000 & a 5-year wait; that doesn't tally up with anyone I know who has received their GRC. Nor are birth certificates issued by the courts; they're issued by the General Register Office. So what you're really looking for here is two separate documents: a Gender Recognition Certificate, followed by a Birth Certificate.
The Gender Recognition Panel is a tribunal that sits at least once a month, and they assess about 16 applications for Gender Recognition Certificates at each meeting. There's an application fee of £140 for the GRC, and it generally takes 14 weeks for them to hear your case and if you're successful, you get a certificate around 2 weeks later. It's very common for them to query your paperwork and to ask for more evidence, so it's worth your while to read up on their requirements and to over-supply if you can. Those queries add extra time to the process so it's good to avoid them.
If your application is successful, the Panel notifies the General Register Office that your gender has changed and they automatically send you a new short birth certificate - free of charge - complete with instructions on how to apply for a full birth certificate. The full birth certificate costs £9.25.
Here's where you apply for a GRC (in particular, have a good read of the guidance leaflet numbered T455 and T451); and
here's a good overview of the process, including more info on birth certificates.
