Long reply - sorry!
RoW law in England pretty much trumps all other laws, including privacy and property laws, human rights law etc..
The group will either probably try to exercise the 'it's been used for years' argument thus it is a de facto and historic RoW, and if there's any older historical evidence to suggest use: eg longstanding personal accounts from before you were owners, they will have a strong case. Or they may - as in our case - use some actual historical evidence, such as an actual order or old map to demonstrate that it exists in actual law. In our case, the malevolent nutter Ramblers Association member, who applied to have the RoW opened used an old map. we were told there was NOTHING we could do, if the application was upheld and could have been forced to allow people to walk through our garden. We were duly refused a mortgage on the basis of this pending application.
We won though by providing alternate historical evidence which showed the route applied for was wrong - ie did not cross our potential garden...
You need specialist legal advice, will try and dig up name of the person I almost used.
Would also suggest if you have time, ask for copy of formal application from the group from your council to see where exactly they are claiming a path...
And then go to your local record office/archive and ask to see any right of way stuff covering that land in their quarter sessions documents
I would also suggest you adopt a conciliatory tone with the walkers and try to compromise and the law is very much on their side however much you feel it is wrong. You can apply for a diversion order if a right of way is established but they have the upper hand...
...and it isn't necessarily wrong that they do, since most private property was 'claimed' by rich landowners from 'common land' many centuries ago and the right to roam freely is a very sensitive political issue and also very complex. Your recent ownership unfortunately doesn't begin to touch the historic rights that were established long ago...
HTH - will post more on this if you want...