Write, pitch, repeat. Working as a runner might help you network and understand the industry, there are various mentorship schemes, but ultimately nobody gives a shit unless you've got a great script. You don't need permission from anyone to write a script and send it to a commissioner.
There's a mile-long queue of people who want to be a script writer, but a much more exclusive club of people who actually write and pitch scripts regularly. As the great Ronnie Coleman said, "everybody wants to be a body builder, but ain't nobody wanna lift no heavy-ass weight".
I was at the BBC for a few years. I can't talk for much other than Children's BBC, but there many of the people involved in that area of production had also worked in a range of roles in the BBC and migrated over to that position. Alternatively they had worked writing and creating outside of the BBC, possibly in passion projects but likely on other paying gigs - possibly with agencies. But generally they were people who were able to point to a large body of previous work.
I don't think I ever saw a role like 'script writer' on the internal job sites.
Don't forget that a large amount of the production is out of house - through production companies of varying sizes.
What I loved about working there was the genuine passion for content and the audience that most people there had - good luck if they can get in.
