I'm in the UK and my referal to the mental health services took around 6-8 weeks to hear from them. The actual appointment was for a month after that (end of May for me). I ws beginning to doubt it, but it did happen. It also depends on your PCT - as each PCT will allocate more/less resources to whatever they decide. So, that means it may take longer or shorter for you to get your screening consultation scheduled.
When you go, be sure to get a general mental health check. Tell them this. "GENERAL MENTAL HEALTH CHECK" - probably best not to yell, but be firm and make sure they don't try and start counselling or whatever. Once they've signed off that you're not looney, then they can refer you to the nearest GiC - and it's the GIC that wants a clean general mental health check.
Then for HRT on the NHS, go full time, do a year of RLE and then they'll consider you for HRT. There are ways around that but I do not condone them, so I'm not listing them. Then after a year of HRT, you'll be eligible for consideration for surgery should you wish to go down that route.
OR, go private. I don't know what the options are in the north east, but London has a few options and will usually follow the SOC rules/guidelines pretty closely. You can always switch back into NHS system without penalty for this. You'll get two appointments on the NHS before you having to commit to the GIC programme if I'm not mistken. They will likely do their own re-assessment of everything and that might piss you off

(one woman I know had her prescription significantly cut back by the NHS when she switched systems - but that's a whole other story in itself)
Private consultation in London may not be so bad if you can make it to London. The intial consultation and first follow-ups may happen relatively quickly - so plan to stick around for a couple of weeks if you can. Normal follow-ups though could be as infrequently as every 3-6 months so there's a lot less travel involved. If you can handle the extra cost, time and travel, it may be worth it - but that's up to you.
I've started both routes, so when/if I ever hear from the GIC in London, I'll have that option available to me too.
Hope this helps