They may have broken UK consumer law here. The CS rep didn't simply say something off the top of their head, they spent 15 minutes checking. However, it would have been better to have insisted that they put this all into an email so that you would have proof that you had been fully verified, as without proof they can simply say there is no record of this chat session, and it would be the player's word against that of the casino. UK consumer law accepts that the UK has no ID card, and the UKGC are expecting operators to use electronic database verification, not documents. No matter what their terms say, this is a case of being "treated unfairly" as a representative from the company not only gave a reassurance that this was not an issue, but took the trouble, and 15 minutes, to check with a superior beforehand.
Given that making the deposit was based on the assurance from the CS rep that verification had been passed, this is also a case of mis selling of the gambling product, which is a breach of consumer law in it's own right. It looks very much like CS at WillHill are behaving in the same manner as bank sales reps in the old days, something that cost the banks an absolute fortune, and still is. In short, CS lied in order to get the deposit, because had they told the truth that they could not accept alternative documents, they would not have generated revenue from this account.
There is no specific requirement in law for ONLY accepting a passport or driving license either, this is purely down to company policy. The law simply requires that "reasonable measures" are put into place to detect and prevent money laundering. A birth certificate can be used, but further accompanying documents are required, more documents in total than would be the case were a passport or driving license used. There is even a government list of "A rated" documents that will suffice for most situations" and a "B list" of alternatives for use when the person being verified does not drive nor travel abroad.
One way to break this deadlock is to take them to the small claims court, where you would demonstrate to the judge that you are who you said you were to WillHill. The judge would do the KYC, and if you pass, WillHill will have to pay up because they are relying on KYC failure for non payment. You may also be able to add some expenses as part of the claim, such as essential costs in preparing for the case, and statutory interest at 8% on the amount of the claim. It's standard practice when a company is taking a consumer to court for not paying a bill to add court costs on top. Many councils slap on a charge of £75 or more when taking council tax non payers to court for a liability order, and provided the cost isn't seen as unreasonable, the courts are allowing it.
A verbal contract is actually binding in law, often an issue when people phone their phone or internet provider and agree to extend their contract purely over the phone. If consumer law is OK with this concept when the business is the "winner", it's going to hold a business to a verbal contract when it's the business trying to wriggle out of it because they have ended up on the "wrong side" of the agreement. CS are agents for the company when dealing with customers in the course of their work, it's a myth that the excuse of "poor training" or "staff incompetence" is a legal means for a company to wriggle out of an agreement made by it's own employees.
.... although it went all wrong this time. 
