No, with a private repair when Mrs Smith has crashed into Mrs Jones, Mrs Smith pays for the repairs to Mrs Jones' car directly. Neither insurance company has to pay anything. Mrs Smith's premiums will probably still go up, because her insurer now sees her as higher risk, but not as much as they would have done if her insurer had paid for repairs. Mrs Jones' premiums usually don't increase. If it goes through insurers, Mrs Jones may see her premiums go up or lose her NCD, even though Mrs Smith caused the accident.
It's only fair that Mrs Smiths premiums don't go up as much as they otherwise would, because she's save her insurance company the cost of repairs. Overall, it's not clear, and goes against economic logic, that Mrs Smith is any better off.
In any case, we don't care about her, it's Mrs Jones that matters.
Mrs Jone's will only lose her NCB if her insurance company doesn't recover costs. In the scenario we're supposed to beleive that the other party can be trusted to pay, so supposedly this is a non-issue.
Mrs Jones will probably have increased premiums, but why would her premiums increase more in scenario (a) no-claim, no cost to her insurance company, than in scenario (b) no-fault claim in which insurance company recovers 100% of costs, so no cost to the insurance company. Either way, her insurance company is out nothing. Either way, her premiums will probably increase.