Health tourism absolutely is a thing. There are even companies that specialize in arranging it. However, my understanding is that it predominantly falls into two categories, neither of which is particularly about exploiting systems like the NHS:

1) People traveling to e.g. the US or Israel to get procedures that aren't available in their home country.

2) People traveling from high-cost to low-cost countries and paying everything out-of-pocket because even forgoing insurance and adding travel costs still leaves it drastically cheaper than their home country.

There are some cases of people scamming government-funded healthcare, but I think those have been sensationalized by the press.