I think it really varies, based on my readings of the long-running thread.
For us, it was fast the first time because we could take pretty much any appointment (DP's work was flexible). They needed to do the paperwork, so that's one appointment, where they'll discuss plans with you. Then they'll do various tests - some to make sure you're both healthy (ridiculously, the non-birth partner will need some blood tests too). They'll probably also test the AMH or ovarian reserve level at the same time. And they'll probably also ask for a scan to see how the fallopian tubes are functioning. Though, some clinics will I believe treat you without that, most won't.
If you're then a good candidate for IUI, which there's no reason you wouldn't be, they'll write you a specific plan and find you some sperm. For us the sperm matching took the longest, though it is a bit silly as there's a shortage of donor sperm and not much choice! I think that took about 10 days.
Then, it's a matter of when your cycle starts, and you can, in theory, crack on.
However, all sorts of things as well as clinic appointment availability can make the process longer.
My DP was straightforward. I went in, and an early scan found I had a large ovarian cyst, so I had to have that treated (which took months) before I could start again. We also had a delay because of sperm availability. You could also find you get into the treatment phase and you don't ovulate that month, or the follicle never gets big enough, so they have to tweak your medications (or, if you're on non-medicated IUI, rethink!).
Basically, before you get to the actual insemination bit, there's an awful lot that can be up in the air, which explains why it took six weeks with my DP and 15 months and counting for me.