The best way forward will be to now send a formal fencing notice which specifies the fence you want. Get a proper tradesman to quote for the costs and this will also help with the wording in the notice. Then give the notice to the neighbours and they can either agree or give a cross notice. The relevant parts of the Fencing Act are below.
10Notice to do work to be given
(1)Any occupier who desires to compel any other occupier under this Act to contribute to the cost of work on a fence shall serve on him a notice in form 1 of Schedule 1 or to the like effect.
(2)The notice shall—
(a)specify the boundary or line of fence, or the parts of the boundary or the line of fence, along which the work is to be done; and
(b)specify (whether by reference to a fence described in Schedule 2 or otherwise) the work proposed to be carried out with sufficient particularity to enable the occupier on whom the notice is served to—
(i)comprehend the nature of the work proposed and the materials to be used; and
(ii)estimate the cost of the work; and
(c)specify the consequences of failure to comply therewith.
(3)If it is proposed that the cost of the work shall be borne otherwise than in equal shares, the notice shall state the shares that are proposed.
(4)Except as provided in section 16, in the absence of agreement to the contrary the occupier of any adjoining land shall not be liable to contribute to the cost of any of the following:
(a)any part of the work on a fence that is done before notice relating to the work has been duly served on him:
(b)any part of the work that is done after the due service of such a notice and before the due service of a cross-notice on the occupier who gave the notice or the expiration of 21 days from the date of the service of the notice, whichever first happens:
(c)any part of the work that is done after the due service of such notice and cross-notice and while differences between the parties remain to be resolved either by agreement or by the court.