It depends very specifically on the situation. Unadopted and private roads and a complex matter. Most roads and many tracks are public in that they are highways or there is a public right of way. However, they are not necessarily maintained at public expense; these are referred to as unadopted or private roads.

The law on the maintenance and adoption of such roads is complex and contained within Sections 203 to 237 of the Highways Act 1980. In respect of the Road Traffic Act and other related offences, these generally come under the principle that a private road is not necessarily a road to which the public does not have access, nor is it exempt from the law. Generally speaking, most of the law is applicable on an adopted road. For example, most of the offences committed under the 1980 Highways Act are designed to punish those who endanger or interfere with users of a ‘highway’ or who damage or obstruct a ‘highway’. A few offences go further in that they apply not just to highways but to all ‘streets’ as defined in the Act.

When it comes to speeding, some offences will be covered by the Road Traffic Act, but again — it's not clear cut. For example, local authorities have powers to regulate traffic in their areas via Traffic Regulation Orders (TROs) as a result of the 1984 Act. The powers are generally exercisable in relation to roads, and again the word ‘road’ being defined in section 142 of the 1984 Act in the same way as in the 1988 Act, as ‘any highway and any other road to which the public has access, and includes bridges over which a road passes’.  There is a general speed limit of 30 mph for ‘restricted roads’, and a road is a restricted road if it has a system of street lighting with lights less than 200 yards apart.

However, the local authority can include or exclude roads from the category of ‘restricted roads’, as it thinks fit. The 200-yard rule is thus not absolute. Apart from this general limit, there is the power to vary speed limits on roads. Hence a private road to which the public has access may or may not have a speed limit, but if it has street lights less than 200 yards apart the 30mph limit is likely to apply.  Local authorities do have general power to put up traffic signs. If a private road is a “restricted road” because of its lighting, a speeding offence may be committed even if there are no signs indicating the speed limit; but if the road is restricted by virtue of a decision by the local authority, no speeding offence can be committed in the absence of signs.

As for further steps to take, I would recommend speaking to your local authority (council), as well as the local police force and potentially the owners of the campsite to make them aware that you're unhappy with the current situation. At least then you'll have done everything in your power to make the road safer.