Hi, Interesting project...
So, I think I understand this:
1. Reed switch / magnet senses water is flowing. (Does this cycle on-off like a rotating magnet, or stay on??)
2. Thermistor senses output water temperature
3. SSR is cycled to control water temperature.
So the Arduino has a pretty simple job:
- IF Water is flowing enable the temperature control (Otherwise make sure no heating happens!)
- IF Temperature control is enabled, monitor temperature with thermistor, cycle SSR appropriately.
(This implies you have a way to set the desired temperature. Can be a simple external Potentiometer that you later calibrate.)
Or "His" and "Her" buttons? (You have to decide what to do if "Both"

)
As in that saying from the 60's "Save Water .. Shower With a Friend".
Later you can control your wood stove...
Let us know how it goes....