Unfortunately, many people do not read the finer print in their insurance policy when they buy the cover. There are numerous reasons why an insurance company will refuse to pay out. They could reject the claim if the driver was not an authorised driver, if the driver was an employee on company business, if the driver was under the influence of alcohol or over the legal limit and even if the accident was the result of the driver's negligent or reckless behaviour.
Another bitter pill to swallow - and I have had complaints from people who experienced this recently - is if you are late or miss a payment on your monthly premium, the insurance company can reject your claim.
What I suggest you do is get a full and detailed assessment from the insurance assessor. If the tyre in question was not the cause of the accident, I am not sure it should be held up as an excuse for non-payment.
If your car was parked in a parking bay outside an office block and a photocopy machine fell out the window onto your car, what would the insurance company do then? Declare that your tyre was bald, therefore no payment?
It does not sound fair and is in fact not fair.
The insurance company needs to come clean with all the facts, all the test results and all the substantiated reasons for refusing your claim.
Also, contact the Ombudsman for the insurance industry and get an opinion on your case.