How can you synchronize sound effects and music with a live radio sketch?

They have mixers, and mixing boards, and with the Technology of today, it is done digitally, of course in Studio, but ya need the equipment.

There are two answers High Tech or Old School. The High Tech way is to have a sound effects man in the both working with your actors using cues from the script at his command a digital sampler with the effects pre-loaded. The one I use has about 2800 effects from running down a foggy alley way to Star Wars Blasters  (hitting a telephone guide cable with a screw driver). Old School is what I used back in the days when edits were done with a Block and razor blade  (I miss Tape!) you have a "BOX O STUFF" You want a punch to the gut, Punch a head of Lettuce in a pillow case, Celery also good for organic fight sounds . a walking platform ( two 4x4 , two 2x12 pieces of wood large enough for 2 people to walk or run in place ) add reverb for location effects  just make sure your sound effects person knows Microphone technique and plans everything ahead of time.