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No claims discount on car insurance?

I’ve been driving in the UK for 13 years and never had a crash or made a claim on my car insurance. However, my wife has had three accidents, all of which have resulted in claims against our joint policy.

I have now separated from her (not as a result of the accidents!) and want to buy a car of my own. When I look for car insurance quotes, can I state that I’ve had 13 years of no claims or do I have to take into account her 3 claims because they were made against a joint policy?

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If the insurance was in your name with your wife as a named driver then it is your insurance that has been claimed against and will need to be declared. If the policy was in your wife’s name with you as named driver then your future insurance policies are not effected. However if the latter is the case then you do not have a no claims discount as you were not the the holder of the insurance.
Some companies will honour the fact that you have never claimed as a named driver on someone else’s policy and give you a small introductory discount on this basis.
Yes you have had three claims but I would tell the insurance company the circumstances. Did you have no claims protection? It may shield you from the worst
You have to be very careful what you declare on an insurance application and make sure you answer every question to the best of your knowledge or your insurance is invalid. I strongly recommend you speak to your insurer and ask the question specifically, different insurers may view your situation differently.

No claims is more commonly becoming more like an accured discount than a benifit for good driving. I suspect they will count the claims against your policy rather than accepting that you have personally had no claims.