There are different types of visa. You say you sponsored them. Do you mean this, or are they here on a spouse visa? The 2 year term & "no recourse to public funds" looks very like a spouse visa. The visa stamp in the passport should state the type.
There is a definition somewhere in legislation of what public funds means (can't find it at the moment), & I don't think (but can't be sure) that it includes council housing. It is more related to welfare benefits.
So if it is a spouse visa & I am right about the definition it should be possible to get council housing, providing you are not deemed to be intentionally homeless. However, if you want Housing or Council Tax Benefit it will have to be assessed for you as a single parent, & there could be a problem if your wife does not have a National Insurance number.
This can all be pretty complex. I suggest you go to your local CAB & take all the relevant documents with you.