You need to specify exactly what you intend to do - literally write it down - from excavations to foundations to wall materials, buttresssing to engineers specification, finish, capping - the lot. This will form help form the gist of the Scope of Works. There will be other items to include but this is the basic intent.
Then take legal advice as to how best to proceed in relation to the provisions of the Land Conveyancing Law Reform Act 2009 Chapter 3:
Interpretation of Chapter 3.
43.— In this Chapter, unless the context otherwise requires—
“adjoining” includes adjacent;
“adjoining owner” means the owner of any estate or interest in a building or unbuilt-on land adjoining that of the building owner;
“building” includes part of a building;
“building owner” means the owner for the time being of any estate or interest in a building or unbuilt-on land who wishes to carry out works to a party structure;
“the court” means the District Court;
“party structure” means any arch, ceiling, ditch, fence, floor, hedge, partition, shrub, tree, wall or other structure which horizontally, vertically or in any other way—
(a) divides adjoining and separately owned buildings, or
(b) is situated at or on or so close to the boundary line between adjoining and separately owned buildings or between such buildings and unbuilt-on lands that it is impossible or not reasonably practical to carry out works to the structure without access to the adjoining building or unbuilt-on land,
and includes any such structure which is—
(i) situated entirely in or on one of the adjoining buildings or unbuilt-on lands, or
(ii) straddles the boundary line between adjoining buildings or between such buildings and unbuilt-on lands and is either co-owned by their respective owners or subject to some division of ownership between them;
“ works ” include—
(a) carrying out works of adjustment, alteration, cutting into or away, decoration, demolition, improvement, lowering, maintenance, raising, renewal, repair, replacement, strengthening or taking down,
(b) cutting, treating or replacing any hedge, tree or shrub,
(c) clearing or filling in ditches,
(d) ascertaining the course of cables, drains, pipes, sewers, wires or other conduits and clearing, renewing, repairing or replacing them,
(e) carrying out inspections, drawing up plans and performing other tasks requisite for, incidental to or consequential on any works falling within paragraphs (a) to (d);
“works order” means an order under section 45 (1).
Rights of building owner.
44.— (1) Subject to subsection (2), a building owner may carry out works to a party structure for the purpose of—
(a) compliance with any statutory provision or any notice or order under such a provision, or
(b) carrying out development which is exempted development or development for which planning permission has been obtained or compliance with any condition attached to such permission, or
(c) preservation of the party structure or of any building or unbuilt-on land of which it forms a part, or
(d) carrying out any other works which—
(i) will not cause substantial damage or inconvenience to the adjoining owner, or
(ii) if they may or will cause such damage or inconvenience, it is nevertheless reasonable to carry them out.
(2) Subject to subsection (3), in exercising any right under subsection (1) the building owner shall—
(a) make good all damage caused to the adjoining owner as a consequence of the works, or reimburse the adjoining owner the reasonable costs and expenses of such making good, and
(b) pay to the adjoining owner—
(i) the reasonable costs of obtaining professional advice with regard to the likely consequences of the works, and
(ii) reasonable compensation for any inconvenience caused by the works.
(3) The building owner may—
(a) claim from the adjoining owner as a contribution to, or deduct from any reimbursement of, the cost and expenses of making good such damage under subsection (2)(a), or
(b) deduct from compensation under subsection (2)(b)(ii),
such sum as will take into account the proportionate use or enjoyment of the party structure which the adjoining owner makes or, it is reasonable to assume, is likely to make.
(4) If—
(a) a building owner fails within a reasonable time to—
(i) make good damage under subsection (2)(a), the adjoining owner may apply to the court for an order requiring the damage to be made good and on such application the court may make such order as it thinks fit, or
(ii) reimburse costs and expenses under subsection (2)(a) or to pay reasonable costs or compensation under subsection (2)(b), the adjoining owner may recover such costs, expenses or compensation as a simple contract debt in a court of competent jurisdiction.
(b) an adjoining owner fails to meet a claim to a contribution under subsection (3)(a), the building owner may recover such contribution as a simple contract debt in a court of competent jurisdiction.
Works orders.
45.— (1) A building owner who is in dispute with an adjoining owner with respect to exercise of rights under section 44 may apply to the court for an order authorising the carrying out of specified works (a “ works order ”).
(2) In determining whether to make a works order and, if one is to be made, what terms and conditions should be attached to it, the court shall have regard to section 44 and may take into account any other circumstances which it considers relevant.
Terms and conditions of works orders.
46.— (1) Subject to subsection (3), a works order shall authorise the carrying out of the works specified, on such terms and conditions (including those necessary to comply with section 44 ) as the court thinks fit in the circumstances of the case.
(2) Without prejudice to the generality of subsection (1), a works order may—
(a) authorise the building owner, and that owner’s agents, employees or servants, to enter on an adjoining owner’s building or unbuilt-on land for any purpose connected with the works,
(b) require the building owner to indemnify or give security to the adjoining owner for damage, costs and expenses caused by or arising from the works or likely so to be caused or to arise.
(3) A works order shall not authorise any permanent interference with, or loss of, any easement of light or other easement or other right relating to a party structure.
Discharge or modification of works orders.
47.— On the application of any person affected by a works order, the court may discharge or modify the order, on such terms and conditions as it thinks fit.
As you can see there are a wide range of options, checks and balances in the matter of party structures now, which may assist you in resolving your problem.
All advice on AAM is remote from the situation and cannot be relied upon as a defence or support - in and of itself - should legal action be taken.
Competent legal and building professionals should be asked to advise in Real Life with rights to inspect and issue reports on the matters at hand.