I'm in a similar position, but as well as a new tv I also moved from BT to Virgin, partly, due to the fact Virgin offers Sky Sports in UHD, which BT doesn't.
I'm no jargon expert by any means, but BT Ultimate on Virgin is 4k (which relates to the number of pixels etc) but is not HDR which relates to the dynamic range of the colours (I think!). Sky Sports UHD on Virgin *is* HDR.
On BT, Ultimate streams on 2 different channels, one is 4k HDR, one is not. I did watch the HDR channel on my new tv prior to moving to Virgin, and although I obviously haven't had the chance to watch them side by side, my layman's observation is that the sharpness of the picture etc is very similar via both providers but perhaps the HDR version via BT may be a touch more 'Vivid'...?
Just my (by no means expert!) impression.