Usually for absolute beginners I'd say book a package holiday with Crystal/Inghams/Nielsen etc. but for a 4-night holiday this won't work as these companies mostly do Saturday-Saturday one-week packages. You can still look at these companies in order to get an idea of which resorts are popular with British guests and then choose a resort or two to research further.
There are a number of specialist travel companies that offer shorter breaks, but these can be expensive and only usually go to the most expensive resorts, which might not be the most suitable for a beginner.
Which leaves organising a trip yourself, which as your partner has already been on ski trips and presumably knows the basics should not be too difficult. Beginning of March should still be fine for most major resorts in France and Austria. Look at where the airlines fly to and work from there - e.g. Geneva for the French Alps, Salzburg or Innsbruck for the Eastern end of the Austrian Alps or Zurich for the western side of Austria.
The previous poster mentioned Zell am See in Austria. I'd recommend the neighbouring resort of Saalbach-Hinterglemm for a first-time skier as there is more for beginners and a far better apres ski vibe. As it is somewhere I know well, this is how a trip to there would work (MNs who usually go to France can maybe do something similar for their favourite French resorts) :
Flight to Salzburg - easyjet, Ryanair and BA all fly from the London airports and there are flights almost every day in the winter.
Salzburg to Saalbach - Private taxi (See www.saalbach.com/en/service/arrival) or by train or bus via Zell am See.
Accommodation - The Tourist Office website (saalbach.com) has an accommodation service - you put in the required dates and this is circulated to the hotels and guest houses etc. Or airbnb/booking.com etc. for hotels and apartments. Tripadvisor for reviews - and look at the hotels that the Package Tour Operators use, as these can also be booked direct or via booking.com
Every second shop in Saalbach rents skis and boots. Lift tickets are available from the lift stations - no photo required. There are several ski schools, all with English-speaking instructors if you need an instructor on the first day (hint - this is a good idea, not only for finding your ski legs without you partner but also for finding out the best areas to go to and the pistes to avoid).