She should register as a resident at latest on 3rd month (because as a tourist you can only stay 90 days). Then she has 3 more months to choose an insurer - so at the end of 6th month she should make up her mind. Or the government will intervene and make a choice for her. Then she'll be asked to pay the bills starting back from when she registered here, i.e. 3rd month.

Yes, probably you'll get the first bills some time in Juli, August or so, if you really need so much time to choose an insurer. But any insurer you take will back-charge you the premiums starting from April. So the first bill you get will be relatively large, covering several months starting from April.

Put it this way - if you arrive and register in April, but don't get your insurance sorted before July you could be looking at a first invoice of around 900-1,200 francs given that your monthly premiums may be 300-400 a month. Better to arrange it relatively quickly after you register. The same applies to your girlfriend when she registers.

You must register when you arrive and before you start work. You have 3 months from that date onwards to find and subscribe to a health insurance provider but the premiums will be backdated to the date you registered so you have nothing to gain by delaying it. If you don't subscribe to health insurance by the end of the 3 months then the canton will assign one to you and they don't usually choose the cheapest option.

For your fiancée the easiest thing for her to do given that she is EU is to come in July as a tourist and start looking for work. It would be trickier to register and get a permit for her as you wouldn't be married.
Once you're married at the end of September and you come back to Switzerland as a married couple she can register and apply for her permit under the family reunification system unless she has already found a job by then. She is allowed to spend 90 days in every six months in Switzerland as a tourist without having to register.