It takes a hell of a long time, although given your age it may be speedier, I was 26 when started TTC so it wasn't felt there was any rush (was 5 years from TTC to baby).
From memory, you could go alone or with your partner. They do need to see you partner at various points along the way to get consents and things but I think the initial appt would be ok just you.
The GP might ask a few general questions, how many cycles TTC, history (have you ever been pregnant, does partner have any children, any signs of endo etc).
Referral time just depends. We were referred to a wonderful but large busy clinic and the wait times are quite long. Think it was around 3 months to see the consultant for the first time. You'll be seen a few times after that for bloods etc and to decide on a treatment plan. We had to have been TTC for 2 years to be referred because of our young age, but as you are over 35 it's possible 6 months is enough (it would be in my area).
Think they asked if we were married but just for the notes. If they are treating a couple I think they want to be treating a stable/long term couple though if you see what I mean, and some places have funding criteria around previous children and things so they need to know your family situation.
The treatment plan will depend on what the issues are. They might think Clomid will help if it's an ovulation issue. They might work out it's a sperm issue and decide that way. We never went down the Clomid or IUI route or anything because it just wasn't suitable for our condition. They might offer you a hysteroscopy though to check for issues.
How long? It really depends. On so many factors. Possibly in your case around 12-18 months from referral to embryo transfer but that's a real shot in the dark. For us it was much longer, we had around a year of investigations which led to nothing, then placed on the IVF waiting list which was a year, then had egg collection but got very ill with hyperstumulation so waited another 3 months for a frozen embryo transfer.
This time round we had frozen embryos and initially approached the clinic in Nov 16, saw consultant in Feb 17, booked in to start frozen cycle in May, cancelled last minute for health reasons and started cycle again in June with frozen embryo transfer in mid-August. And that was paying privately...
My advice is just to go to GP to get the ball rolling. It won't do any harm to get things moving and know where you might stand. Have you been TTC with OPKs etc?