Lobbying v. grassroots advocacy
This is just semantic but I think you probably don’t want to call what you’re proposing a “lobbying group.” Lobbying usually refers to one particular form of advocacy (face to face meetings with legislators) and in many countries it is regulated more heavily than other forms of advocacy.
(It’s possible that in the UK, “lobbying group” means something more general but in the U.S.)
 This is true in the U.S., which I know best. Wikipedia suggests it’s true in the EU but appears less true in the UK.
Who else is working on this?
Here are a couple small examples of things being done along these lines, though I agree there is little overall:
-Resolve to Save Lives claims to do some advocacy for epidemic preparedness in low-income countries in collaboration with the Global Health Advocacy Incubator. The latter group seems to be hiring an Advocacy Director though the posting is old so I wouldn’t be surprised if it’s out of date.
-PATH has done some advocacy to encourage the U.S. government to invest in global health security.
This is very useful, thank you!
Yes lobbying prevents charity / nonprofit registration in the USA, but advocacy doesn’t.