Hello Claire. You only need to buy top up health insurance if you are earning more than the threshold earnings level . this is €11,657 for a couple.
There is a national benefit which all low wage earners in France who have lived here for 3 months or more need to use and is called La CMU complémentaire. CMU stands for Couverture Mutuelle Universal and is the name for 100percent health cover
CMU complementaire is means tested, so even if you earn more than the amount above, you will still be entitled to a partial "couverture", right up until about 17,000 € of earnings. After Tax.
here is the link to all the info about CMU. You will get cover from the day you apply if you are eligible and it's absoloutely vital if you want to avoid huge hospital, gp and prescription bills for anything. French healthcare is better than in the UK generally and you are more welcome to argue and shop around for a second opinion I've discovered.
If you think you may need CMU, you ask the RSI once you have registered as an auto-entrepreneur, the RSI being the organism which oversees healthcover for self employed people. I think it would be the CPAM if you use the "cheque a la personne" system of working.
Plummers are "Artisans" and your partner will get help from your nearest 'Chambre des Metiers et des Artisanats' (Hope I spelled that right). and you also have to register at the Chambre de Commerce to become an auto-entrepreneur. In fact I think you have to go to the chambre de commerce first.
It's not hard to register, and doesn't cost a lot, I think it was 15 euro for the card we needed as commerçants, but be prepared for fun and frolics with paperwork, especially from the RSI and read everything you sign at the Chambre of Commerce carefully, even after signing, because they made two errors with us and we've been waiting 3 years for permission to trade.
good luck and learn as much French as possible. My tip is to watch French news and then try to follow the subtitles on a simple slapstick style comedy each night. I recommend "scenes de menage". Hilarious.