The closest BBC Radio 4 gets to a 'jingle' is the hourly 'time-signal' or 'beeps'...except at 10am on sunday mornings when it's replaced with the Archers' omnibus music intro....(the only time, I think).
I can take or leave jingles. Believe it or not they are very expensive to make and until quite recently were still produced on 8 track where they were played by DJ's on banks of players.
Knowing one or two local BBC Radio station managers I was always fortunate in getting my hands on discarded players which we used in our hospital radio station and for that I was thankful. I'm not too sure how they are done these days. Some DJ's had very good ones but other were pretty crass.
My main beef is that the BBC seems to have forgotten that radio 4 is not designed to be listened to by the mentally challenged, it's something people grow into as they mature and as such has a constantly growing market without seeking younger listeners. It is, of course, necessary for Radio 4 listeners to accept that this years 40 year old's grew up with different tastes and triggers to their predecessors.