Late Enrolment - Can I still start this September?

Hi, I have received an offer from LSE for my master's course which I am really excited about. I am currently studying History at the University of Bristol. Due to issues with my health, I have been unable to submit my dissertation which means that I will have to submit it in August during the re-assessment period. This will mean that instead of graduating in August, I will graduate in November this year. This is not a big problem, but I am worried about how this will affect my entry to LSE this year. It is very important I begin my Master's this year in September 2023 as I cannot defer and take a year out (I was forced to take a year out of uni last year due to health concerns, however the lack of structure and isolation meant that I went through a severe depressive episode for over 6 months and had psychotic symptoms which caused me to be hospitalised several times. I only got better end of Sept 2022 which meant that the start of my final year was very rocky and is part of the reason why I was unable to submit my diss. But I do not want to go through anything similar again and must avoid taking another year out at all costs).

On the LSE website, this is the information they give:

If your degree will not be awarded until after the enrolment period, you must supply an official document from your university stating your final marks, the date when your degree will be awarded and confirming both that you have met all the requirements to obtain your award and that the current grade provided will not fall below your degree condition

I was just wondering if anyone else has been in a similar situation and have enrolled at LSE while not having their official degree document, and whether they had any issues with their own uni providing these documents/ LSE accepting them ?

Any help, information or advice would be greatly appreciated.

Thank you so much.