Loryn..if the council can fob you off, they will (I know, I used to work for one). Genuinely homeless with children IS a priority. It may not be as high a priority as, let me think, disabled with disabled children homeless and bankrupt, but it's still high.
Someone who is being evicted and cannot find private rented (usualy because private rented turn them down), esp on low income, should stay put, get evicted and harrass the crap out of the council on the grounds that they are going to be on the streets with kids. Giving proof of low income and the efforts to find their own accom, with proof of being turned down. The council may put you in temp accom - if someone was REALLY REALLY without other options, they will accept this, if they don't and get private rented instead, then they always had other choices and weren't the high need, iyswim.
Sometimes, until they have a house available so they can give a homeless offer -they put folks in b&b, hostel etc (worked like this when I worked for the council,don't know if they still do this) in the short term .... our local council now also has a few houses that they keep strictly for people waiting for a permanant home - so you could go in there, but when a house came up you have to move, you don't have the same right to stay permanantly If they accept you as homeless (again this was how it was when I worked there) you got one offer (and it was usually the stuff nobody else would accept) and if you turned it down they took you off the list. But it was something and if you are facing actual on-the-streets-tonight homelessness, you're not going to be picky!
When I worked for the council, folks threatening to go over the road to the local paper offices usually got helped fast. Funny that, isn't it.