My understanding of the legislation regarding homelessness is as follows:
If you are deemed to be unintentionally homeless and in priority need (because of having children in this case) then the Housing Dept have a duty to re-house you. Having said that all councils are desperately short of housing stock (most of it is sub contracted to HAs) but there is still a massive shortfall with demand far exceeding supply. But the duty remains, and so they would have to find accommodation. They are not meant to put families in B & B (and not for longer than 6 weeks) but they do this in some areas of the country where there are no other options.
Because of the shortage of council/HA properties LA Housing Depts are now in the ludicrous situation where they have to house people in private rented accommodation in order to fulfil their duty, but tenants can claim Housing Benefit if they fulfil the criteria. If you are unfortunate enough to live in London, you could be housed anywhere in the country, where the rents are cheapest.
NB Don't vote Tory
Incidentally even if you are deemed to be intentionally homeless but in priority need they have a duty to provide you with temporary accommodation.
LA Housing Depts can loan advanced rent and deposits etc on a private rented property as this is a cheaper option for them than having to provide accommodation themselves.
OP PLEASE stop worrying about social services - (retired social worker here with 30 years experience) and child protection procedures - it won't happen - there has to be evidence that a child is being significantly harmed. Time was when Social Services would be able to provide some financial help for families in need of housing (and yes S.17 of the CA 1989) is still in existence but they too are severely cash strapped so it's highly unlikely you would get any financial help from them. Sometimes social services and housing work together but this will be on the basis of helping homeless families not invoking child protection procedures (unless of course there was evidence that a child was being abused/neglected)
I agree with others that SHELTER (the biggest Housing Charity) are excellent - have a look at their website, and you can talk to them on the phone.
I'm wondering OP about this "help to buy" scheme (not that I know anything about it) and there may be a catch but might b worth a look at the details. The other thing is there are properties where you can buy a half share (again worth looking into) there were some really nice houses in our local paper being offered for sale on this basis (3 bed semi) for around £60/£70 K - it's a really tough time for families and anyone in need of housing as owners can keep increasing rents and evicting tenants if they think they can get higher rents.
CAB might be a good place to give you information. The other excellent charity on all welfare rights is Child Poverty Action Group (CPAG)