Hi,
I'm sorry that you're having to cope with this as a new mum. Firstly, your partner had no right to "kick you out", as you have a child together. You can see a solicitor - first half hour usually free. You need a family solicitor and you can find one in your area from the Law society of England & Wales' website:
http://www.lawsociety.org.uk/choosingandusing/findasolicitor.law
Secondly, moving in with your parents - when you were evicted from your partner's home, you were "homeless", which gave you rights. You'll probably need your parents to say that they have agreed that you can only stay there temporarily, say, for a month in order to retain your homeless status. Contact your council and ask if they have a tenancy support worker service. Tenancy support workers are separate from housing, but help you with practical matters and liaise with housing officers with you.
Visit your local housing office, and remember to say that you are "homeless". Local authorities have an obligation to re-home homeless people and, as you have mental health problems (depression) and a baby, you should be deemed a 'priority'. If you are declared 'homeless' by the local authority (although staying at your parents' house may affect this), then you should not go on the normal waiting list, but on the homeless register. You should also contact local housing associations, telling them that you're homeless too. Hopefully, you will be re-housed quite quickly, but the time varies in different regions.
I don't know what your financial circumstances are, but you will also be entitled to a range of benefits for you and the baby as a single parent. I know it's a lot to take in, but as soon as you start sorting out the practical matters, you'll start to feel more in control.
Good luck and keep us updated with how you get on.
Love,
Wendy
xxx