I think the offer is really great – if I were you, I would go for it. I agree with those here who say that such a chance probably only comes once.
The bathroom is certainly not my taste either – but it’s quite cheap to renovate. Lay a design floor over the tiles, maybe replace the toilet – that already makes a big difference. You can always completely redo it later if it bothers you in the long run. But first, it’s all about saving money.
How about the kitchen? Did I miss that? Can you use the existing one?
Regarding your expenses: From now on, allocate fixed budgets and make sure you stick to them. You get 60 euros per month for your gaming. Not a cent more. Your wife gets 50 euros for nails. 100 euros per month for eating out/takeaway. Savings for insurance/taxes/car. Savings for gifts. It’s best to open savings accounts for the savings portions and fund them by standing order, so the money is "gone" at the beginning of the month and not tempting to spend. Sit down and just set budgets.
Of course, you have to restrict yourselves; but if you really want the house and it means quality of life for you, then in my opinion that outweighs expensive vacations and new cars. The only problem is if, due to the mortgage and high other costs, you can no longer afford a new washing machine, etc. So definitely keep an emergency fund for that.
But absolutely start immediately not fulfilling every wish of the children right away! That not only costs money – but even more later on! – it also doesn’t really make the kids happy. Think about how you as a child enjoyed gifts, how you looked forward to birthdays or Christmas and your wishes. If everything is always fulfilled immediately, the anticipation disappears. And also the satisfaction. It’s sometimes hard, I know – but I witnessed it with my nephew, for whom eventually even the insanely expensive (purchased in installments) notebook was no longer a big deal at Christmas.