Hi OP.
Congratulations on the pregnancy (I assume it's happy news?) 
How about the father? Will he be contributing?
Where do you work? Depending on how much CMS you get, I'd say you're probably best off moving to a cheaper borough, or outside London altogether.
I assuming you don't have an urgent need to stay in Greenwich, or even London at all?
It would be different if your family were close by. Quite understandably you'd have more need to stay close for support (and perhaps childcare).
I wouldn't rely on the council unfortunately.
London is the epicentre of the public health housing and homelessness emergency.
Governments going back 30-40 years have levelled London's housing right down.
Seriously OP, two thirds of England's homeless families are in London. You likely wouldn't want to be where they put you.
It might be different if it wasn't London - because the other regions have more affordable housing (than London) for the low to medium waged and more social housing.
But in London(because governments don't care about Londoners) whole families often have to live, sometimes for several years, in rooms in really grotty temporary accommodation.
Social housing waits in London can be 10/15 years or more.
It's horrendous.