When I was doing rescue I saw one case where spay incontinence seemed to start immediately after the surgery. She was an adult dog, and I say "seemed" because she'd only been in our care a couple of days before the surgery. Other than that, the earliest I ever saw was before the dog was two, so say within a year to 18 months after the surgery. Every other case I've seen or heard about developed it in their senior years.
None of the dogs I've known with it had swollen stomach or excessive thirst. They just were incontinent. The most obvious sign was urine spreading behind them when they were sound asleep. That can also be a sign of a UTI, though.
If your vet can't come up with a diagnosis, consider getting a second opinion.