If your wife's pay would have grown following, as a minimum, the rate of inflation; you would have a lot more cash than you have now.
Furthermore, her job would undoubtedly be easier than it is as she and her colleagues would be backed with better resources and staff without the stress and strain which is on the NHS. The situation that the family have found themselves in, is indicative of Tory neglect, whether or not they have 2 kids or 4 kids.
A family with both parents working should not be kicked onto the streets or bouncing between emergency accommodations, as a result of them having 4 children.
Depends what your willingness for risk is I suppose. The old adage is that you shouldn't quit your job unless you have up to six months of living expenses saved up. For me, it's the same when you're considering something that is undoubtedly going to be pretty expensive. If you're hand to mouth every month then that leaves you one unforeseen event from the crapper for you and your family. Is it a good idea to do that expensive thing?
I don't share your optimism that a 11% pay rise would fix the NHS' problems one bit, but sure I'm sure the wife wouldn't turn it down. Would we want to rely on whomever the government of the day is to bail us out? Heck no. They're tossers.
Regarding your last point, I'm not sure tbh. The average cost of raising a child is said to be £160,000, so if you have four of them you're looking at an annual cost of £35,000 a year over 18 years. The average household income in the UK is £37,000 a year. Even if between you you're on £70,000 a year, after tax you would have about £1,000 left each month to pay your own expenses.