There is no general "Right to Roam" in England and Wales, in contrast to Scotland.
The Countryside and Rights of Way Act 2000 only allows public access that can be recognised and "Right to Roam" on "Access Land".
Access land is defined as such:
(1)In this Part “access land” means any land which—
- (a)is shown as open country on a map in conclusive form issued by the appropriate countryside body for the purposes of this Part,
- (b)is shown on such a map as registered common land,
- (c)is registered common land in any area outside Inner London for which no such map relating to registered common land has been issued,
- (d)is situated more than 600 metres above sea level in any area for which no such map relating to open country has been issued, or
- (e)is dedicated for the purposes of this Part under section 16,
but does not (in any of those cases) include excepted land or land which is treated by section 15(1) as being accessible to the public apart from this Act.
While not the legally definitive map, to most intents and purposes and UK's Ordnance Survey Landranger (no. 199) and Explorer (no. 124) maps can be considered as good as you are going to get information on the UK. It is possible to view limited areas of Ordnance Survey mapping use Microsoft's Bing Maps, such as the area around the Battlefield. (Notice that "Great Wood", immediately to the east is marked as access land). You can also view the maps directly from the OS, however it requires an account and either a subscription or use of a limited trial). However, looking at this area, there is no indication that the land is either common land or open access land, and the elevation above sealevel is 20-25m. There are also no public rights of way indicated along this route. Natural England also keep a map specifically for access land, that is available here
Given that there is a closed (and locked) gate, at the point of access, it would seem reasonable to suggest that one crossing past the gate could realise they were not invited, and be found to be committing trespass, although it would be difficult to demonstrate any loss as a result of such.
In addition to this point, it would appear that English Heritage both charge for and restrict access to the battlefield. As such, they might not be happy for you to access the site from this route. If you have the tickets, then it is unlikely they could claim trespass, but you might get told off.
In practice, you are likely able to make this journey without attracting any particular attention (at least as far as the private track goes, not so sure about walking on to the battlefield). However, if you were to regularly take groups through this route, the likelihood of you eventually being observed and challenged, or the landowner noticing large numbers of footprints and better securing the access increases.