OP - You said that the mother would ben 'entitled' to a cheap quarter and childcare. The forces do not provide childcare, so I dont know where you get that particular piece of nonsense.
Yes quarter prices are generally cheap - but not as cheap as single accn. If mother and father are paying a mortgage on their own home, the cheapest option is likely to be the best.
Depending on where mother is working, what rank she is, what trade she is etc she could well be out of her 'bed' from 0600 to 2000. She may also have to do 24 hr duties where she could be called out in the middle of the night (so childcare??). She may be at a unit where she would get a week long duty (even more childcare??). She could be at a unit where she has to go on Exercise a lot - for weeks at a time (even more childcare??).
As she is in the Forces, every year she has served she has been working towards a regular pension and lump-sum gratuity that will make hers and her childs lives much easier once the military contract is fulfilled. It offers stability in an ever unstable world. The mother is earning a very good, very secure wage, which again in these times is a godsend. I mean, for example if mother was a soldier with 18 years under her belt she could either work for another 4 years (even if that means working 'away') and gets a lump sum (which could be as much as £50K) plus an immediate pension (of maybe £1K a month - that increases at age 55) or she leaves now and gets a lump sum of £8.5K and no pension until age 65, and then only at a reduced rate.
Which would you choose?
Oh and unless the grandparents actually told the OP that they didn't want to do the childcare, then how do we know that they didn't jump at the chance and relish the opportunity?
So.
I don't think the mother is being unreasonable. I think she is probably making the best of a bad lot and working hard to secure a future for her family.
I think I dont know enough about the father to have an opinion (did he tell the OP he 'couldn't be bothered to look after his kid' or did the OP make that up / assume that?). On the face of it he's a tosser, but who knows?
I think the OP is being VERY unreasonable to judge the mother so harshly just because she has made choices that the OP wouldn't have. Especially without the full facts.