Thank you all so much for your replies.
I plotted everything into DNA painter, including the 6th match that I had and the dates that I know. My numbers came out much smaller than yours, rsel, but it still had my mum being the daughter of a half sibling of person one as most likely relationship (3,429), followed by grandchild of a full sibling of person 1 (313) - which can't be the right relationship because I know that person 1 and my mum's mum definitely had different mums.
I was surprised to find that the hypothesis that my mum is a half first cousin of persons 3, 5 and 6 scored only 1, given that their CM numbers didn't seem drastically different to persons 2 and 4 on the other branch.
The third scenario you guys suggested, that my mum is the grandchild of one of person 1's other uncles (other than the father of persons 3, 5 and 6) came out at a score of 101.
With these scores, would you say it is very highly likely that Person 1's father is my mum's grandfather and that all the other options are very unlikely?