Speaking as a reasonably politically engaged Brit, I can safely say that the main reason that there isn't much interest from the UK is that most people wouldn't have heard of this initiative.
I would conjecture this is for two reasons.
Firstly, and I appreciate it's important to you but, the UK's engagement is somewhat dominated by Brexit currently, so fairly peripheral initiatives, or even Bills, are not going to get much attention.
Secondly, the proposal is pretty lame. It's basically suggesting that the EU says some nice things about minorities and throws a bit of money in their direction. In the UK, politically engaged linguistic and national minorities have, by and large, a lot more autonomy/power than that. I'm thinking the Scots, Welsh and Irish here. There are some exceptions but they're either not numerically large e.g. the Cornish or their languages/cultures aren't particularly threatened e.g. Hindi speakers. Where we do have significant issues, this proposal isn't going to get close to addressing them. I'm thinking about the prevalent attitude to Islam here.
As a consequence of these, the politicians and journalists who would normally bring these things to the public's attention are unlikely to feel it's worth the effort. Essentially, there are bigger fish to fry as it's unlikely to have any material effect on our cultural or linguistic diversity. Or address any of our actual issues.
Now, I appreciate that you may argue that, even if it's not that important to the UK, we have a lot of people who could help bring it to fruition for others for whom it is important. And I have much sympathy with that position. But the timing sucks unfortunately.