I am a senior practitioner (caseworker) in homelessness and assess homeless applications. So, I actually assess applications when someone applies as homeless. I would like to add to what Cuddles wrote.
If someone approached me and was 6 weeks pregnant and renting a room I would say you are not currently homeless within 28 days. The reason being is that the baby has not yet been born and the accommodation is currently suitable and available to you. However, if you have other complications, as Cuddles wrote, the accommodation would have to be assessed as to whether it is reasonable or not.
There are two avenues into council housing - the homelessness route (which is what I work in) or the housing register. I always explain it like this to my clients: Homelessness is if you are going to be homeless within 28 days, already homeless or in accommodation that is not suitable (e.g. severe overcrowding). The housing register is when you are already in accommodation, but not facing immediate homelessness. However, if you do go on the housing register, you can apply for more points depending on your circumstances. Some people do go on the housing register when they are in unsuitable housing and are allocated extra points in order to be housed faster.
Also, a 1 bedroom flat for a person with a baby is considered to be reasonable since the livingroom can be used as a bedroom and you can be granted a permanent tenancy for this. Babies are usually counted as 1/2 a person.
I agree with being proactive with your application. When you apply for either route, it is always good to have the following items:
ID (passport, home office document, birth certificate - original only)
Proof of income (proof of benefits, wages - last 3 month payslips, last 6 weeks payslips)
Any documents with your addresses on them for the last 5 years
Your 5 year address history in chronological order
Antenatal notes
The above applies for both you and your partner.
The more information provided, the easier it is for the caseworker to investigate your application. The guideline is that a homelessness case should be completed in 33 WORKING days (not 33 days from the application, but take into account it is 5 working days to a week), but it is only a guideline and not a rule.
However, I wish to add that your caseworker will not have any power over how fast you get housed once you are accepted no matter how much you hound them. Most councils are now on a bidding basis and it is up to the person to take on the responsibility to bid for properties.