Summer 2017 may be a tad optimistic.
Wait times very much depend on which GIC you go to, and what process they'll expect you to follow when you get there. You mention Coventry and Durham: well, the nearest GIC to Coventry is Daventry, and it has the shortest waiting list in the country... so I hope you've been referred there. As of today, their waiting list for a first appointment is approximately 10 months. As for Durham, the closest one is Newcastle but it has a much longer waiting list of well over a year for the first appointment.
Most GICs have a set pathway that you have to follow: in many cases, you'll need at least 3 therapist appointments (each at least 3 months apart) before they'll let you see their doctor... even if you've already been diagnosed privately. This is especially true if you're very young, as you are: genital surgery is irreversible so they'll most likely be extra-cautious. It's unlikely that the doctor will refer you for surgery at your first appointment with them; they usually like to get to know you over a couple of appointments before doing that. Those appointments, of course, tend to be several months apart.
They generally want you to have been on T for two years before you have bottom surgery; this is so that you can get maximal growth of your 'junk' before surgery. Bottom surgery and top surgery are completely separate procedures and neither one is dependent on the other, so it won't make any difference if you don't need top surgery.
When the doctor agrees that you need a referral for bottom surgery, they'll need to send you to a different doctor for a second opinion before they can refer you for surgery. Often, this takes place at a local hospital, and you'll usually see a psychiatrist (or similar) who will ask you the same sort of things the GIC doctor asked you, just to confirm your diagnosis of GD and to give their approval for surgery. Then they'll write to the GIC, and your GIC doctor will then refer you for bottom surgery.
Bottom surgery is performed mainly in London by Mr Christopher's team. After they get the referral from the GIC they'll be in touch for you to have an initial consultation; this might take 3 months or so. Once you've had that, they will slot you in for surgery somewhere in the region of 3-6 months later. And that's just the 1st stage.
So best guesstimate of your expected timeline? And these are educated guesses because I don't know your case but I'm familiar with the system:
Referral to Daventry (10 months): first therapist appointment December 2016
Last therapist appointment (and this is a wild guess!): late spring/early summer 2017
First GIC Doctor appointment: +- Sept 2017
GIC referral for second opinion before phallo: if you're lucky, Sept 2017; otherwise probably late 2017/early 2018.
Second opinion appointment: early 2018.
Referral to phallo team: mid-2018
1st Stage phallo: late summer/early winter 2018.
As for your Uni, I'd recommend ringing their Student Services departments and asking that question before you apply. You don't have to mention what type of surgery you're having, but you will have to tell them how much time you'll expect to take off for your recovery.
