How this plays out is going to depend heavily on what your Passport country is. And the sort of medical care that you need. If you can clarify this then that will help people to give better advice.
If you are from the UK, on a UK passport, then it should have been no problem to get a permit, provided that your employer has done the correct process. Whilst you may not have the physical passport, you should have gone to the town hall within two weeks of arriving and registered, and if you ask they will give you a 'registration paper' that confirms that you are registered, and then you can use that to get the health insurance.
The problem is, I think a lot of us are suspicious that your employer hasn't done the right thing to register you properly.
If you are non-EU then very likely your employer has employed you illegally as they have not followed the correct process (Non-EU au pairs need to be employed via a registered agency and the permit has to be obtained before you arrive). There is *No Way* a Non-EU person can arrive and work without first getting a permit.
Do you know that you are actually registered at the town hall now ? Or is it just what your employer is saying ? Are you living with the host family in their home ?
Your country may have a reciprocal health care arrangement with Switzerland, but if it's not urgent medical needs, then you are expected to go home for health treatment.
According to this au pair agency, it is normal for the host family to pay half the health insurance. https://www.aupairworld.com/en/au_pa...am/switzerland
Do you know that you are 'actually' registered with the town hall and that the work permit is a formality. If so, you should know which town hall you went to (you go there yourself) and you can go there, ask them to help you understand your status, and find out if you are actually permitted to be working and living in Switzerland whilst you wait for the permit. If you are legitimate, then you can tell them you are having trouble getting the health insurance and if they can issue you a paper permit (might cost a bit of money, in Zurich they cost us 30chf each)... so that you can get the health insurance sorted.
According to this link, the health insurance is paid by your host family, this is normal for 'au pairs' - so the cost of another insurer, one that is less difficult to sign up with - might be not a problem, in fact, it could be your employer's problem - they need to make sure you get insured.
Personally, I'd be very suspicious that your employer either doesn't understand the system at all, or is deliberately misleading you.
Normally the insurance is retro-active, so be aware that when the insurance does come through, it may be back-dated to the date of entry, and they will make you pay for the whole period - is your employer paying for this, taking it from your salary, or waiting until it actually gets sorted out ?