It won’t happen. To be a legally enforceable fine you’d have to break a law, and it seems in the New Forest biking off legal routes is prohibited under local by-laws rather than the CRoW / Wildlife & Countryside Acts.
So OK you’d be fined, and unless you volunteered your identity, in order to fine someone you’d need their name and address (possibly DOB I forget) and only a policeman has the authority to demand this – council employees don’t have the powers to ask this of you, they may ask / intimidate, but you don’t have to say anything.
Like fatsimon mk2 said, basically you’d have to stand around, wait for a copper to turn up in the middle of nowhere (which they won’t) who at worse would ask you not to do it again as they really couldn’t be bothered to turn it into a case. And if it ever got that far you could fight it in court, ask for evidence to support the fine / it wasnt me guv / there aren’t any signs / etc. And its bad PR really.
In fairness the FC couldn’t really send a press release out and say “we cant really do anything about this” so are bluffing. I do it all the time 😛
By the look of it royal parks are policed by .. the police and / or park rangers have the powers to issue fines.