If you need to have your Spouse visa early, I don't know where you are now, but you can get married where you are and bring your country's marriage certificate (and some other papers too I believe) with you to Japan for immediate reporting to the Japanese authorities, then apply for Spouse visa promptly, or you can even report the marriage outside Japan to the Embassy of Japan in that country. Or you could also go the path of applying for Certificate of Eligibility, sot that you will be given Spouse visa when you come to Japan. Please check the different ways to apply for Spouse visa.
Once you apply and the papers are in order, it should not take longer than a few weeks/months, but this depends on how busy Immigration is and other factors, so I don't think you can get a firm answer here or from Immigration (and I don't think you can hurry them).
But even prior to actually obtaining the Spouse visa, once the two of you are married and have reported your marriage to the local city hall in Japan, have an address (not a hotel) in Japan, she and you can register yourselves at the city/ward hall (she for resident registration, and you for alien registration), and also can ask to apply for the "national health insurance."
It is true that normally non-Japanese become eligible to join the National Health Insurance scheme when you have done the Alien Registration provided the period of residence granted upon entry in Japan is one year or longer.
However, even if your residence status upon entry is less than one year (assuming you come on a Temporary Visitor Status, you get 90 days at the airport as an American), if you are expecting to live in Japan for one year or longer, which you should be able to show to the city hall by telling them that you are married to a Japanese, you become eligible to join the Naitonal Health Insurance program, and also you "must" sign up for it anyway.
To have PD covered by insurance, you need to sign up for the physical handicapp certificate ("shintaishougaisha techou") and specified
ailment/treatment registration ("tokutei shippei ryouyou juryou shou") as well, as it seems. Please have your fiancee look into this in advance (the procedure for these registrations may differ from one prefecture to another).