We are trying to register a car we have taken out to France. Two complications. I have failed to access the ANTS site and attempts to get a personal space for myself via France Connect have failed. We had one initial response asking for documents to which we replied but have not received any response since despite reminders. There is a further problems however. The car was registered in my wifes name in the UK and the tax clearance certificate has also been issued in her name. However our numero fiscale is in my name and she does not have her own. Is there any way to register the car in her name without a numero fiscale.
I don’t believe a non resident can register vehicle in any EU state.
Legally, no, they can’t.
Michael, welcome to the forum.
As a non-resident… you are not supposed to have a French Immatriculated car here in France… you can bring your UK car over and back… but that is it 
Others will chime in, as this has been thrashed about for years…
Last year, prior to me having a numero fiscale, I used this web site to import / register my car in France:
https://www.cartegrise.com/france/demande-carte-grise-standard?dem=6&iframecode=FrenchProperty
You need to be aware that cartegrise will not return your V5.
Once the process is complete, the site has a link to order new number plates.
Is this new? I have registered two previous cars without problems pre the on line system.
France’s public information system is now more “joined up” in that tax / social security / vehicle registration / etc etc etc records are now integrated, so the administration computer can automatically cross check and link all an individual’s administrative records rather than than handle each process in isolation.
The curse of systems integration, the loopholes are all closing. 
A holiday home client here brought over a RHD car and was able to register it without a problem, this was four years ago so maybe things have changed.
I think ANTS went live late 2018/early 2019? It’s quite recent.
It was definitely before 2018, ironically after all the effort and expense they never use the car and it just sits on the drive.