'There is no way we could afford private rents here & as a working household we need to live within commutable distance'
But you also said you were wiling to move out of London
.
What gives? You had one child in the one bed, but instead of focusing on getting better accommodation under your own steam, you paid privately to have another. And then elected to not use contraception so you have another.
Now you want a 3 bed. In London. But you are in Band C.
Good luck with that.
The homeless and those who find themselves disabled and unable to physically access their property will always come way before you, and they will never stop coming.
So you need to start considering some alternatives besides social housing in London in a 3 bed place.
I hate the Tories as much as anyone, but they did not bring this situation on you.
'We are desperate !! so willing to move out of London if given the opportunity.'
No one is going to give you an opportunity. You need to make one.