You really can feel like a fish out of water!!
If your GP has organised bloods for you and sperm analysis for your OH, you will be able to get those results before you're actually seen by the hospital. In my experience, these shouldn't take more than 2 weeks to be available and it is ok to chase the GP for them. It will help manage your expectations as you will have an idea as to whether there are any initial problems. No news is generally good news though! I found it very helpful to ask for printed copies of results. We had a poor experience with NHS fertility at our first clinic and I just felt so much more informed by having results written down so I could research a bit further.
Once the fertility clinic has your results, they can discuss your next steps. This would probably be more investigations or treatment depending on what the results show.
In my area, the GP also booked an ultrasound scan before referring us to fertility. If this hasn't been mentioned, I would expect that to happen after you're seen at the hospital. Things vary a lot depending on area.
Hope that helps!