A few months ago I was part of the tower team involved in the situation OP asked about. A VFR aircraft (with a clearance) took off from an area shielded from ATC radar. The clearance should have resulted in it being seperated from the final approach. When first observed on our radar display it popped up on final, going the other way, indicating a climb through 1000ft. There was a B737 on the ILS at 12 o'clock, 3nm, descending through about 1700ft. The surprised aerodrome controller reacted in what seemed like nanoseconds, and as the Boeing pilot acknowledged the climb/MAPP instruction she also reported "TCAS climb". I found that fairly reassuring from the viewpoint that it would have prevented a (likely) collision, had no climb instruction been issued.
My friends on area control tell me that TA's are fairly common, and RA's used to happen in the situation where an aircraft is about to level off with someone above/below coming the other way, probably because of the much higher closure rate. They seldom seem to happen near the aerodrome. Think I've seen about 1 or 2 a year, and only 1 total where there was an actual loss of seperation.