Firstly, congrats to your cousin. Secondly, it's lovely that you're looking out for her.
I can tell you about my experience.
Started IVF 2 decades ago. Underwent MANY cycles using my own eggs which all failed. (Chemical pregs, didn't stick, cycles cancelled, miscarriage....the lot)
The ONLY pregnancy that went to full term was when I was 3 months away from turning 40 was by donor egg (& sperm owing to DH having issues too).
As with all of these scenarios, it is 50:50. Every step of IVF as well as a 'natural' pregnancy is 50/50 at proving ti be productive.
The bonus for your cousin is that the egg would have come from a younger woman & been screened for a hell of lot of diseases etc & been given a clean bill of health. This is the best you can ask for.
If, as your cousin is saying, she is a few weeks pregnant, I can only imagine that her chances of reaching full term are the same as any woman. She has overcome the hardest challenges.
Hopefully you can put the worries aside & enjoy this much longed for & wanted child.