On the assumption like most, if not all, local government owned land this plot is properly registered, then a successful claim for a ten-year adverse possession will depend on whether or not the council (or others) opposes the claim. If this opposition is successful then a subsequent twelve-year adverse possession may succeed.
See Schedule 6 of the Land Registration Act 2002, which is summarised by the HM Land Registry's Practice Guide 4:
after 10 years’ adverse possession, the squatter will be entitled to apply to be registered as proprietor in place of the registered proprietor of the land
on such an application being made the registered proprietor (and certain other persons interested in the land) will be notified and given the opportunity to oppose the application
if the application is not opposed (by ‘opposed’ we mean that a counter notice is served; see Giving counter notice to the registrar in response to notice. Instead, or at the same time, the registered proprietor may object to the application on the ground that there has not been the necessary 10 years’ adverse possession; see Objecting to the squatter’s application for the implications of such an objection.), the squatter will be registered as proprietor in place of the registered proprietor of the land
if the application is opposed, it will be rejected unless either
it would be unconscionable because of an equity by estoppel for the registered proprietor to seek to dispossess the squatter and the squatter ought in the circumstances to be registered as proprietor
the squatter is for some other reason entitled to be registered as proprietor
the squatter has been in adverse possession of land adjacent to their own under the mistaken but reasonable belief that they are the owner of it, the exact line of the boundary with this adjacent land has not been determined and the estate to which the application relates was registered more than a year prior to the date of the application.
However, if it is opposed then all may not be lost...
- in the event that the application is rejected but the squatter remains in adverse possession for a further 2 years, they will then be able, subject to certain exceptions, to reapply to be registered as proprietor and this time will be so registered whether or not anyone opposes the application
The "certain exceptions" are at paragraph 6(2) of the Schedule:
...a person may not make an application under this paragraph (i.e. a 10 + 2 years claim) if—
(a)he is a defendant in proceedings which involve asserting a right to possession of the land,
(b)judgment for possession of the land has been given against him in the last two years, or
(c)he has been evicted from the land pursuant to a judgment for possession.
Whether "simply securing an existing boundary" is enough will depend, as is often the case, on the particular circumstances to establish factual possession:
Factual possession signifies an appropriate degree of physical control. It must be a single and [exclusive] possession, though there can be a single possession exercised on behalf of several persons jointly. Thus an owner of land and a person intruding on that land without his consent cannot both be in possession of the land at the same time. The question what acts constitute a sufficient degree of exclusive physical control must depend on the circumstances, in particular the nature of the land and the manner in which land of that nature is commonly used or enjoyed … Everything must depend on the particular circumstances, but broadly, I think what must be shown as constituting factual possession is that the alleged possessor has been dealing with the land in question as an occupying owner might have been expected to deal with it and that no one else has done so. (para 2.1 of Practice Guide 4)