You should first obtain an estimate of repairs.
If neither your neighbour nor the contractors admit to this then I would recommend that you send both of them a formal letter before action to demand payment (your estimate will confirm the amount) within 14 days and say that if they do not pay you, you will issue county court proceedings against them. See attached template as an example of what to say. Please let me know if you cannot see the letter which I have uploaded. You will need to tailor this letter to your situation.
You may also want to threaten a report to the Trading Standards (against the builders).
You will need to register at http://www.moneyclaim.gov.uk so that you are ready to issue the claim in the event they dispute the claim and do not pay you. The website is very user-friendly and you would not need a lawyer to use the money claim site. Claims with a value of under £10,000 are classed as a "small claim", so legal costs are not recoverable and the matter may be dealt with on paper by a Judge, not at a hearing. A hearing may be necessary if the court thinks that oral evidence is required to dispose of the case.
You would claim the sum for the loss, the court issue fee (details of the money claim fees are listed here at page 5: http://www.gov.uk/make-court-claim-for-money/court-fees) and court interest which is 8% calculated on a daily rate from the date of loss to date of court judgment. The site allows you to calculate the interest and add it to the claim.
If you are on a low income or have low savings (or in receipt of benefits), you can ask the court for a fee remission so you do not have to pay the court issue fee. If this applies, let me know as there is a form to fill in.
If you win then once you have the CCJ from the court the defendant has 14 days to pay in full. If they do not then it gets registered with the credit agencies after 30 days. You can also enforce the CCJ with the county court bailiffs or transfer the debt to the High Court for a small additional fee assuming the total amount owed is at least £600 and you can use the high court enforcement officers who have greater powers than county court bailiffs. The transfer fee is added on to the debt and payable by the defendant.
There are other enforcement methods which I can help with, including bank account freeze, charging order on their property (and then apply to force a sale), apply to summons them to court for questioning, all of which can ensure you are actually repaid the money.
You may find they just pay you after receiving the letter before action – hopefully they will want to avoid litigation. One of them would be found to be liable - so the one found liable would have to pay the innocent part's defence costs. I would think the neighbour will place blame on the contracts in this situation based on what you have said.
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