I used to work for a UK private health insurance company. For them (at the time) and for most of their competitors, the general policy was they would could (and usually would) exclude you for pre-existing conditions. So high blood pressure, diabetes, etc. However, some companies had contracts with us where there were no exclusions at all. So if you entered with a pre-existing thyroid condition, you'd get coverage for it if and until it became chronic.
So it really well may depend upon your employers contract with the insurer.
As for weight, I can't recall exactly, but I seem to remember that at the time, there were no weight exclusions... but that may have changed and may differ from insurer to insurer. Still, they would only have exclused weight-related issues (and yes, that's a fuzzy area). So if you broke your leg in a car accident, that would be covered.
Aside from private insurance helping you jump the queue and getting private rooms, many of them also have nice coverage for things like physical therapy, chiro, osteos, etc. that can be hard or impossible to get on the NHS.
Feel free to ask me more if you'd like though like I said, it will largely depend upon your employers contract (their HR people will have the details). And also, it's been a while since I've worked in the industry.
If you know who the insurer is, they generally also have quite extensive info online.
HTH.