Let’s analyze this information. Question: did @MiS talk to someone in English?
There are two kinds of services at certain hospitals. One kind is for common folk, the cost is paid by NHI. You present your card and pay down a «co payment» when you make an appointment, which usually includes medication and tests. For example, at my local hospital a visit to the cardiologist is 350 NTD and I get a 3 month refill prescription. Appointments can be done through phone, computer or app.
The second option is that in certain hospitals they have International Medicine Department. For one whole hour, a specialist of your choice and upon request is all yours, attending to your needs in your own language. Booking requires at least 10.000 NTD.
In either case, waiting time depends on doctor availability. Unless the doctor is in a sabbatical, I do not see how can it be a year. With NHI you can choose any doctor at any participating clinic or hospital.
Now by law as soon as you have ARC, meaning legal residence, you will start having your NHI deducted from your salary, and you will get a nice card with your picture on it. That should happen within 3 months of you working here, otherwise NHI will not be happy. They are very tight about collecting their share.
So unless you will be on probation a whole year, and hence working illegally without ARC, then you cannot use NHI, will not be covered and should pay full treatment out of pocket. Not highly probable, though.
I think there was some miscommunication and they quoted you some weird stuff from their International Medicine Department.
Anyways, you talked to one hospital. And one that puts a lot of airs for the service it gives. We have hundreds here. And for NHI matters, you should ask NHI. Yes, we know that if you call in English no one will answer. It would be more recommended to visit any NHI office, of which there are also hundreds, and try to sort stuff in person. Do not expect replies by mail, especially in English. Follow @tando 's link.
As stated, no one waits a year for NHI, previous conditions do not have anything to do with it, rather work visa/residency issues matter more