I guess you need to replace the connector on the ESC. I think you can buy a pigtail (so you can either cut the existing wires and solder the pigtail onto them, or take off the original connector and wire and solder on the new pigtail with connector.
Here would be a test I would do first. Take off the prop. Start the system up, and as it is running, wiggle the wire and/or the connector near where it makes contact to the timer. If the motor stops, then it is the connection.
Another possibility is the the wire prongs on the timer don't make a good contact to the timer circuit board solder pads. I had a JMP-2 which arrived DOA from the store. I was just about ready to send it back when I looked more carefully at the circuit board and saw that a couple of pins were missing solder, or didn't have a nice clean solder connection on the circuit board pad. A small touch of the soldering iron and a little solder fixed that up, and that's the timer I am still using today.