If she crashes and a claim is made, they will find out that she is in fact in the second or third year and therefore driving illegally, which invalidates her insurance.
Only partially. The insurance company cannot refuse to meet a third-party claim just on the basis that the person was doing something illegal at the time. If you think about it, a large number of accidents and resulting claims are due to some illegal activity, e.g. running a red light or failing to yield right-of-way when required. So despite clauses about the need to comply with the law in all respects for the policy to be valid, they cannot deny a third-party claim, and as that is all that is required by law you could not be charged with driving uninsured in such a situation.
I've seen similar instances debated in the past in a police legal forum which relate to "fully U.K." drivers. For example, if somebody is stopped for no road tax or found to have no MoT certificate, he cannot also be prosecuted for driving uninsured become of those transgressions of the law.
Other parts of the insurance are a different matter, so they could certainly refuse to pay out for damage to your own car under a comprehensive policy, for example. But as those are not legal requirements, that's just a matter between you and the insurer.
The legal issue would be, effectively, driving without a license, or if you held a U.K. provisional too, the charge of driving otherwise than in accordance with a license (e.g. no L-plates, no supervising driver etc.).
Also, does anyone know if the "25 to drive on someone else's insurance" applies to a US license?
It will depend entirely upon the wording of the policy. If it doesn't specifically state that those "other drivers" must hold a U.K. license, then a U.S. license is fine.
Edit for typos.