Cheaper premiums under wife's name but will my no claims count afterwards?

From the looks of it, it seems that you're putting your wife as the main driver (policyholder) and you'll be going on the car insurance policy as a additional driver. If you're going to use the car more than your wife, you will have to become the policyholder, otherwise this will be regarded as fronting and in the event of a claim, you will have your insurance void. Now assuming that you will be an occasional driver, then what you are suggesting is fine.

As a named driver, you do not earn any no claims bonus (NCB); only the policyholder gets NCB and in your case, it will be your wife. Some companies like Direct Line allows the second driver/named driver to earn no claims bonus of their own and they can use it when they take out a policy of their own with the same insurer. So for example, if you were to be insured with Direct Line and after 2 years, take a policy with them but this time you're no longer a named driver but the main driver instead, you will be eligible a discount proportionate to the no claims you hold with them as a named driver before. However if you were to change insurance company and go with say Churchill, the no claims would not count.