Your savings have nothing to do with it unless you want to claim benefits to help pay your rent.
What you are eligible for depends on where you live and the properties available. I'm in Wolverhampton, here you would be allowed to bid on flats or houses with 2 or 3 bedrooms, this is because most of the property is houses with 3 bedrooms but due to the bedroom tax one and two bedroomed properties are most popular.
I was at the bottom of the council waiting list but due to mobility issues I was able to apply for a HA bungalow.
I was already in a privately rented ground floor flat, so adequately housed, but as I know my mobility would reduce further I was eligible for where I am now.
A few of my neighbours moved up from London, their council did some sort of deal with Wolves a few years ago, so people swapped a flat in London for a house with at least the same number of bedrooms.