Is this panel on an outbuilding, not physically connected to the main house? If so, and if it was installed longer than I think 10 years ago, the 3-wire feed and bonded neutral/ground would have been code compliant at the time and still legal. If more recent, current code would have 4 big wires coming from the main panel and the neutral/ground separated (and if it's a subpanel in the same building the code back then would have too).
Which specific handle cut the power to the pump -- the 2nd from the left that has a red wire coming from it (the leftmost double breaker, the right handle of that breaker)? And which handle cuts power to the other red wire -- I'm guessing the rightmost handle of the rightmost double breaker just from the wire color shade?
That's definitely wrong for a 240V setup, as both of the red wires come from the same leg of the incoming feed (and even if they weren't, 240V breaker handles must be tied together). Properly supplying 240V would require a tandem breaker like this (just an example, need one that's listed for the specific panel):
https://www.homedepot.com/p/Square-D-Homeline-2-20-Amp-Single-Pole-1-20-Amp-2-Pole-Quad-Tandem-Circuit-Breaker-HOMT2020220CP/100186179. The middle two handles and terminals would supply the pump, and the outer two would be independent 120V circuits so there'd still be enough circuits.
However, changing breakers like this is beyond my DIY comfort level especially for a pool area, and doing so legally might require permits and worst case whatever's needed to get a 4th wire from the main panel if has to be updated to new code. An electrician would be a must to do anything further there.
Since it does appear to be a 20 Amp circuit with 12 gauge wire, the simplest answer is to just get the T101 timer as you mentioned and make the whole pump setup 120V. From the wiring diagram at
https://www.intermatic.com/-/media/inRiver/10162-9164.ashx/T101-Instructions-EN, you'd wire it as such:
Ground: Same as current timer
Terminal A (neutral): Connected to the silver and white wires (neutral coming in, and neutral going to the pump). You can put both wires under the same screw.
Terminal 1 (line): Connected to one of the inbound red wires that's connected to the current timer. Cap the other inbound red wire with a wire nut, and turn off/mark the corresponding breaker in the panel.
Terminal 2 (load): Connected to the blue wire going to the pump
Editing to add -- I agree with others that having an electrician that has worked with pools come take a look would be a wise and safe thing to do.
