Oh my, I am so confused...
Let's for the sake of simplicity assume all runs are done on flat ground under same environmental conditions.
As pace increases VO2 will increase although VO2 will plateau or top out at some intensity. (That would be VO2max; you do not achieve VO2max on an easy run.)
Heart rate under moderate conditions is going to increase pretty close to linearly with VO2 (and thus with pace). HR will also tend to plateau at near maximal pace/intensity.
Under some conditions your HR even at the same pace/intensity will rise over time (cardiac drift), but so will VO2.
Heart rate though in real life is tricky. It is not the best metric for intensity as a lot of things can affect it: sleep or lack of, temperature, stressors, caffeine, dehydration, etc.
The watches are estimated oxygen consumption probably from speed and HR data. The issue is that since the algorithms used are proprietary it is difficult to assess how reliable or valid they are. Some researchers may have tested them against metabolic carts, but I have not looked for those studies in several years. There are some that have looked at the validity of things like Fitbits. A google scholar search should turn those up.