We are not insurance experts so our advice is general. In view of the risks you would be wise to consider taking out your own "non-negligence / "all risks" Third Party Liability insurance and if your home above the basement is "finished" (rather than a building site) it would be a good idea to cover damage to your own property too. You should obtain advice from an Insurance expert.
The Contractor may not like it but you should have a qualified engineer inspect the works during the initial excavation and thereafter as regularly as necessary to ensure that the work is being carried out in accordance with the scheme design and the Temporary Works Design & Method Statement. A Party Wall Award made by a competent Surveyor will require this.
Some contractors will take risks with their operatives lives and with your property to maximise profits. Regular inspections by an independent engineer will deter this.
Some damage is extremely likely both to your property and to any neighbouring property if you are underpinning a Party Wall. Your Contractor may exclude liability for what they may describe as "inevitable" or "unavoidable" damage, that is damage that does not arise from negligence or error. This can allow a contractor or their insurers to avoid payment for damage. Under the Party Wall Act you (not your contractor) will be liable for ALL damage caused to an Adjoining Owner's property by the works and related losses (which can include loss of rent or the cost of necessary alternative accommodation), so you should make sure that you have a cast-iron contract with your contractor defining liability for any damage that may be Awarded under the Party Wall Act.
If your neighbours have previously built a basement and you use their basement wall then, under the provisions of Section 11(11) of the Act you will have to pay them a contribution of half the cost.
Watson Woods Partnership
Section 12 of the Party Wall Act gives neighbours he right to require that someone undertaking work deposits a sum of money on an escrow account for the duration of the risk. The money would be accessible only by the Party Wall Surveyors and could be used for repairing damage or completing works that are abandoned (if this is necessary to protect the neighbour's property). Where basements are concerned the sum of money required can be very significant dependant upon the risks
Your basement should be designed by a qualified Engineer. The design of a basement is a relatively straight-forward matter but it is important that the design Engineer visits the site (to appraise the building and any adjoining structures) and does not just work from designer's drawings. Problems are most likely be caused by the method of execution or the "Temporary Works Design".
Many Engineers do not like to prepare Temporary works designs. They assert that it is the contractors responsibility to decide how to construct the design. This is a dangerous scenario. The temporary works cannot be left solely to the contractor as there is a conflict of interest. The safest and best methodology may protract the works or otherwise reduce profit. If the contractor produces a method statement it must be checked and approved by a qualified Engineer. Better still is for you to directly engage an Engineer to prepare the Temporary Works Design & Method Statement.
A Party Wall Surveyor will want to see that a Temporary Works Design & Method Statement has been prepared and will want to include it in a Party Wall Award.
Because of the risks many Party Wall Surveyors like to engage "checking engineers" to review the design information and you will end up paying for the additional fees. A Temporary Works Design & Method Statement prepared or approved by an Engineer can help deter reference to another "Checking Engineer",
If your concrete underpinning is not accurate to within normal building tolerances and if, when a neighbour excacavates for a basement, they have to trim off projecting concrete they can claim the cost arising from you. Costs can add up to several thousand pounds plus professional fees. This can be discovered years after your basement is built but the liability will remain.
It is important that the location of your underpinning relative to the boundary is defined in a Party Wall Award and also that the acceptable tolerance is agreed (we currently consider that +/- 50mm is reasonable). It is then important that the contractor agrees (as part of the contract) to adhere to the above AND accept liability for any costs that may arise in the future if a claim is made because the tolerances have been exceeded. For obvious reasons it is best to engage an established basement company who are likely to be around several years into the future.