There are some very affluent areas of Sheffield, it has some of the wealthiest areas of the UK. However it does have a good range of prices down to comparatively low and some areas are not particularly desirable but your budget is what it is. If you're used to Swiss prices, you'll probably find UK property prices and the general cost of living quite cheap, but that obviously has to be balanced against lower salaries.
What do you plan to do for work? You might need to think about commuting time and costs.
If you're planning on providing daily care for MIL you need to be nearby, or else you could spend hours each day traveling and not be close by in an emergency.
Your DH must know the city and have an idea about where you could afford to live, more than a bunch of strangers who generally have high expectations about the location they would like to live and the budget to facilitate it.
Does MIL own property that can be sold and contribute towards the cost of a home for everyone? If you want a family home with a self contained unit for MIL you will seriously limit choice but they do exist, they are generally called granny annexes here but you can search for key words on right move so you could try searching for the word granny to cover all bases. But sometimes these count as a separate dwelling so will be subject to its own council tax, which is at least £1k pa, plus heating and water etc.
If you don't have the budget for this budget it might be better to look for a larger standard property where she can have her own room on the ground floor if necessary.
You could set up a bit of a kitchenette in her room if she wants to be able to make herself hot drinks or anything without using the main kitchen.
Another point that you should probably consider is the longevity of this arrangement. I hate to say it, but you have to consider that this might not be a long term arrangement, of course she could live another 20 years, but otherwise, what would your plans be after she dies? Buying and selling property attracts costs, which would be lost if you decide to move again quickly, so maybe you should consider renting where she currently lives in the first instance?