Can you post the model number for the VS pump? Many have their own timer an are meant to be continuously powered. and a few have an aux relay that can be used for a SWG. If that is the case ignore the rest of this post for the moment.
You could switch the SWG to 120 V but there would be not benefit in doing so. I would recommend that you open the cover on the SWG box and confirm the wiring is 240V against the drawing in the manual. Also check that there are no other wires besides the Red, Black and Green we can see coming from the timer.

Also check your timer model I think you have this one:
https://dtrermj9469s.cloudfront.net/userfiles/images/inriver/original/42502_p1353me instructions.pdf
Assuming this is all correct and we want to continue to use the Intermatic to control the pump schedule:
For the fix you're going to need a short (maybe 5-6") bit of 12 Gauge black wire and a wire-nut suitable for 3 #12 wires. There are two options, the "minimal way" or the "best way".
Minimal way:
Turn off both the pump circuit breaker and the lights circuit breaker, and if you have tester available verify that the connections on the timer are dead.
Remove the black wire going to the pump from terminal 1
Connect one end of your bit of black wire to terminal 1 (where the wire to the pump was)
Remove the black wire going to the SWG from Terminal 7
Use the wire nut to connect the three black wires together (Terminal 1, Black to pump, Black to SWG).
Remove the Red wire going to the SWG from Terminal 8 and connect it to Terminal 4 (in addition to the red wire going to the pump).
Turn the breaker breakers back on, and now the SWG should come on when the pump does.
Best way:
We will re-program the timer to use it's "Mode 5" this will dedicate circuits 1 and 2 to work together to switch both legs to the pump and SWG and Circuit 3 will become the timer circuit for the lights:

Turn off the pump and ligjht circuit breakers, and if you have tester available verify that the connections on the timer are dead.
Remove the Black and Red wires going to the SWG from terminals 7 & 8.
Disconnect the red wire from terminal 6 (light switched leg) and connect it to terminal 8.
Disconnect the black wire from terminal 5 (light power) and connect to terminal 7.
Disconnect the black wire going to the pump from terminal 1.
Connect a short 12 gauge black wire from Terminal 1 to Terminal 5.
Connect the black wires for both the pump and the SWG to Terminal 6
Connect the red wire for the SWG to terminal 4 (where the red wire for the pump is connected).
Turn the breakers back on and following the directions in the timer manual set the timer to "Mode 5".