My wife and I are currently in the middle of IUI treatment so I can give you my experiences! I'll preface it with saying that as we're in Scotland we get 6 rounds of IUI on the NHS (and will then have the same IVF entitlement as straight couples if they don't work), so this will be from the perspective of someone doing it through the NHS and private clinics may be different.
We were referred in late September, and the first appointment we had was mid-November. This was via video chat as our clinic is 100 miles away from where we live, and here they went through basic health checks for the carrying partner, checked I had an up-to-date smear test, did a visual check of BMI, explained the next steps in the process and answered any questions we had. After this, we were referred to implications counselling, as we will be using donor sperm and this is a requirement of HFEA, and we were put on the list for a HyCoSy.
I had the HyCoSy last week and they were happy with everything they saw there. At this appointment they also did my height and weight to get an actual BMI, and this came out fine too so we're now on the waiting list for sperm donation and when we reach the top it will be IUI.
As we are NHS we have to fit the NHS referral criteria, so those are mostly the things that could have caused them to say no, as far as I know. So those are things like BMI has to be under 30, you can't smoke, they told me to stop drinking entirely, there are age limits. If you're private these may be different than NHS so it's worth checking those out if they have them online.
Hope it all goes well for you and your partner!