It's just a shitty thing to do, to agree to undertake a responsibility that affects someone's child and career and then back out last minute, because sister's partner has to go into work for the day.
Reading through all OP posts suggests the mother is a nutcase and probably not someone I'd want around my child on a regular basis.
Really not sure why OP thought mum would change with the arrival of a baby. Babies are hard work and a lot of commitment, responsibility, energy and time.
OP is also a slightly naive nutter if she had a rose-tinted view of "new grandma" regarding babysitting with delight. She obviously hasn't learned from her own lifetime of experience growing up.
Hint. Having a baby is not going to turn your mother into a different person. She is what she is.
Since the interview did not go well, I'd use the extra free time to attend a mother and toddlers group, make friends with other mums and start building a support network.
Hint 2. The week without fiance would have been so much easier if you were meeting up with other new mums and just sharing being a new mum with people in a similar position.
Time for a reality check - and put a non-family structure in place.
Because fiance's family are also either too distant or not able to or not inclined enough to offer support. Because at 6 months, baby is old enough to spend overnight with grandparents, aunts and uncles, if they're appropriately responsible.