With the death of The Show What You Wrote there really aren't any other true open-sub radio opportunities beyond Newsjack.
The usual advice is to contact the producers of other shows and see if you can get an in there. In my (limited) experience this can work but I'd recommend having a few credits up your sleeve to be able to show you are in some way competent. You may also need to send in a sample of your work in the style of the show but best to limit what you send in initial contact.
The places that are usually looking for writers are topical-ish shows and sketch shows. They don't have to be on the radio you can also chance your arm with TV shows as well although that nut is harder to crack! Find a show you're interested in, dig out the name of the producers (Google usually comes up trumps here) and then send them a polite e-mail explaining why you'd be a good person to send stuff to them in line with their show. I'd recommend not doing a generic round robin to everyone but try to show you've spent the effort on them - after all you're asking them to put the effort into giving you a chance. For BBC Radio producers' e-mails follow a similar form and if you've ever got the Newsjack you'll know what that is. (If you haven't got the NJ e-mail might be hard to get an in anyway).
Another good way to get an in is to get to the recordings and try to speak to those involved in show after. That can be hard but see if you can get an in with a writer you know, cast member or being brave in the pub.
DbM gives some good examples and here's a few I'd recommend trying if you've got some open-sub success:
The News Quiz / Now Show - Usually like you to have proved yourself on Newsjack and often the 'additional material' slots are invited by the producer. But worth asking if you've got previous topical credits. I found all this happens way earlier than you might expect probably if you contact during the current series they'll be starting to fill up the slots for the next one.
Dead Ringers - Again possible to submit. Bill Dare (DR producer) gives some good advice on the Sitcom Geek podcast with him so that's worth a listen. (He recommends the get to recordings approach).
Breaking the News (BBC Radio Scotland) - The closest thing to a next step up from Newsjack. There's a group of writers but they are usually willing to add new names to that list each series.
Comedy Unit / BBC Scotland - BBC Scotland occasionally have other comedy shows (e.g. Wired News), although they seem to be more focused on Breaking the News now. Worth keeping an eye out as they sometimes have opportunities. Again worth getting an in with the producers
Gifted & Gallus - BBC Scotland are also moving online with this new series of filmed sketches (as are others). Worth seeing about getting involved with this as producers work on other programs as well.
NewZoids / Tracey Breaks the News - TV topical shows. Harder to get into but worth a go. Again get in with the producers and see what they want. And even though it's topical they may line up sketches well in advance to give them a reserve to have ready for the run.
I feel I'm repeating myself a lot here but the advice of getting in with the producers and people involved in the show is key. Anything you can do to help that is going to be useful. Of course the key thing is to write funny stuff for them!