As a young whippersnapper you should have good bone density (if you've got a normal healthy background) and that amount of treadmill running sounds very low to be the cause of a stress fracture. You've not suffered any acute injury to the foot (dropped a weight on it) have you?
Going to A+E for an X-ray is likely to be inconclusive as stress fractures only tend to show up when they've started healing. MRI is reckoned (around these parts) to be the best way to spot, but if you've got to go to your GP for a referral you could be waiting some time.
If it's not getting better on its own you should probably seek out some real life person rather than a random bunch of electrons on a webpage. Some people are lucky to have GPs with a sporting background, who know what they're on about, the rest of us have to try to filter through the throng of other people who should know to try and find someone trustworthy.
I am struggling to fit my exercise in around work and as a result I am gaining weight
From what you've said the mornings are realistically the only chance to exercise, accept the fact you'll be slower (body still waking up, slightly fasted state, blood at its thickest) and work on minimising the excuses. Running kit at the bottom of the bed (I've known people to sleep in their kit to minimise the excuses on this one), shoes by the door with the hi-viz, head torch etc. If you're finding hills hard work plot your flattest course.
I'm going to offer a slightly different PoV to SCR. If your budget can run to having multiple running shoes then go for it. I've got 4 pairs of my current preferred training shoe in various forms of knackerdness- by rotating them round you give the cushioning more time to recover (and dry out and become less smelly) between runs. It also gives you an opportunity to break in new shoes before the big day.
In the next three months you're going to be putting quite a bit more mileage into your old shoes, with more chance of them degrading- if you can afford it, spend some time finding their eventual replacements now.