So, here's where I think your problem lies:
When putting the motors in reverse, they draw so much current, that your battery voltage drops significantly. This results in the UNO loosing power and reseting.
And since you don't have breaks on your wheels, reverse breaking is really the only option I can think of right now.
A possible solution:
Try a (rather) large capacitor connected to the 5V and GND of the UNO. That will buffer the UNO's supply, while the battery voltage drops. You could put the cap on the battery, but I suspect you'll have better results with buffering the UNO's supply.
You could break the motors by shorting the terminals, but as far as I've seen, the L298 doesn't support that and adding that feature is more work than 1 cap.
Hope I was of some help.
Greetings
Tom