I've submitted once or twice, never got anywhere though.
Actually, not strictly true now I think about it: one piece they
declined, but asked to see other stuff I'd written. Which didn't
get anywhere. But it made me happy at the time.
Generally you have to suggest the slot. All have slightly different
requirements in terms of length, content, style, and if you don't
comply
absolutely, they won't even finish reading.
Consider that they have five radio stations (is it?), and four TV
stations, all broadcasting round the clock, with a myriad slots to
fill from here to infinity; they're busy. You need to help them out
by pointing at a suitable home for your work, really you do.
A better option might be to go for one of their 'opportunities'
which are posted on their writers room website. These crop up
regularly and call for particular things at particular times -
sometimes even radio sketches.
Their website is a mine of useful information anyway - not only the
'opportunities', but the tips, who to submit to, how to submit,
schedules, which shows are currently recruiting writers and
material.
You can also view their script archive which is fascinating if
you've seen/heard a particular episode of something - you can see
how it panned out from page to screen/ear.
See here:
www.bbc.co.uk/writersroom
Radio is one writing area that I believe is hungry for new work yet
is so often overlooked. It's very cheap to make radio drama, and
lots of slots for it.