Welcome and hope you enjoy time in Edinburgh. +1 that a rat bike is the way to go for cycling to & from University, shopping etc. A road bike would be best kept inside your accommodation and only brought out for the long rides.
Sadly, I don't think the Allez would last long if left out in the streets, and I reckon you can get better value: the University has a useful web page on bike options here:
https://www.ed.ac.uk/transport/cycling/getting-a-bike/second-hand-bikes
If you are used to cycling in London, then potholes, broken glass, pollution and bad driving will all be familiar. Differences to consider (as other people have said already):
1. It rains a hell of a lot more in Edinburgh than in London - you'll get soaked regularly. Mudguards make a big difference in keeping a rider and bike in good condition.
2. Cobbled streets are still plentiful. They are uncomfortable - will rattle you to the point of headache -
and are pitiless in the wet. Like black ice, you can be on the ground before you know anything about it.
3. Edinburgh is hillier than London. That's good for fitness, but may be an initial shock if you've been cycling in London (outside of Hampstead & Crystal Palace).
4. The tram lines & safety.
5. There are some dedicated cycle paths but of varying quality - when I was there, they were plagued with broken glass and I've been stoned by local yobs. Road.cc reported last week that there are still problems:
https://road.cc/content/news/dundee-cyclists-arm-broken-after-teen-jumps-out-him-274545
Don't let any of this put you off - there's a lot of good cycling not too far out of the city. Definitely take the Innerleithen road over the Moorfoot hills, and out through Gifford into the Lammermuir hills.
The Water of Leith Walkway is good too: but offroad & it can get muddy (did we say it rains a lot?).
One of the best circuits is around Salisbury Crags/ Arthur's Seat, which is handily round the corner from the main University campus.
All the best!