First of all you need to appreciate that the next few months will be tough for both of you. There is a time in the pregnancy when your partner wont even like your smell.
It is possible to get housing benefit when you rent one property (it doesn't need to be a council property) and own another. Whether you will get it will depend on a number of factors. If the flat is not being rented out, the council will look at the equity in the flat and, if it is large, this will stop you getting housing benefit. If the equity is small or negative, then you may be able to make a joint claim for housing benefit. If the property is to be rented, then your rental income will be considered instead of the equity when deciding what to pay.
It is possible that your partner will be able to claim housing benefit for her own rented property. As you are not living together at the moment, this could be the first option to explore. However, if you started to live together, you would need to make a joint claim which would involve looking at the flat. It is likely that the claim would involve the council asking questions about your relationship.
If you continue to live separately, you should make separate claims. You would be expected to pay child maintenance and your partner can claim child tax credits and child benefit after the child is born. If you decide to live together, you should notify the council and the tax credits office and make a joint claim.
Ideally, you should be able to gather some figures together, put these to the local housing benefits office and get an answer now about exactly what help will be available. Hopefully you will get some sort of answer but you wont get a definite figure until the claims are made.
There may be a shortage of council properties in your area and the waiting list may be long. However, if you are sufficiently confident about receiving adequate benefits, you can rent privately.
Hopefully you will be able to find work in the next few weeks and a lot of the issues that you are worrying about will not arise.