Sorry this is so long.
Agree with connecting with Family Nurse Partnership (aka FNP) invaluable resource.
Also Family Lives (used to be parent partnership) who have a helpline but it can be pot luck getting through. IME once you do they're a huge resource. 0808 800 2222 Monday to Friday 9am to 9pm. They also have a parenting resource forum.
I'm assuming the father is either not around or also school age. (the later- some last the course, but most don't.)
Gingerbread and Shelter are both excellent organizations, the first (along with family lives) for individual tailored support, both, for knowing the legal position of things especially housing rights vs responsibilities.
People are correct about the realities of doing everything alone at a young age vs potential support, but this may depend on your relationship with your family. Not everyone's are positive.
Others are being kind, I'm going for hard honesty.
An awful lot has changed since I had my first children younger than you without practical support, but while some things have changed on the surface, from YP I know today, a lot remains the same under the surface in terms of what really matters. Some of this may be stressful to read, but it's some honest if hard advice.
Without knowing the fuller details of the situation:
Your situation is legal, (though your parent/s may soon be legally overcrowding in housing terms) and you are now in an odd situation where you are considered a child legally, but are now expected to have the same abilities as an adult, but when like every adult you do something dumb, you will be seen as potentially incapable through age and questions raised, not just a new parent making the same mistakes as every mum ever.
Everyone around you will have an opinion, some of them will also have agendas.
You also may find issues over NEET. Not in Employment or Education, depending on how your LEA treats under 18 pregnancy. Remember whatever their attitude, you have a right to education, don't be encouraged to let it go, but seek what is right for you, not being squeezed into what's easiest for them. There are colleges with baby care on site, remote learning, and all sorts of possibilities if you push for them. Most educators will back you to the hilt if you show willing.
I don't wish to make a difficult situation harder, but you do need to very aware how vulnerable your situation is. It's scary, but being aware will make you a better mum, and help you avoid some situations, and will you get past this stage unscathed.
Your generation is generally more savy about much but many put too much faith in 'rights.' Remember it is always your child who's rights will be being assessed, not yours. Responsibility is now with but also on, you. (all this is useful when seeking your educational rights.)
When we are young we often feel ready and able for anything, as we get older we realize how little we and others, really know. The confidence is very useful, and can carry us through, but it can also blind us.
Many things will depend on what services are available in your area, and how stretched they are.
I'm giving you scenario's based on councils under stress, (most) not comfortably off with lower issues. (fewer) Some areas have great voluntary organizations, and different deals, others have little but overloaded council. Research your area.
Housing services at your age are likely to come through childrens services who may not have the experience needed for it to be positive for you, and are pretty likely to seek a mother and baby hostel placement to discharge their legal housing obligations, for the 1st 1-2 years.
Generally they aren't great solutions for a 1st time 16yr old mum with a newborn. You will be likely to get a social worker which can be a blessing or a nightmare. They are likely to change quite frequently, contact can range from sporadic to overbearing. You will however eventually get housing through this, but it can be years of 'watch and wait' before you get your own place.
Many of these placements are private landlords just charging an arm and a leg to the council for providing accommodation and a (random) person on a reception desk. It can be isolating, and who else is there can be again a blessing, or very unhelpful and lead to interactions with bad situations.
So that's why I'd say if your Mum has your back generally, I'd stay where you are for now. It is considered 'a protective factor,' something very important when a very young mum being assessed by services.
Space issues will actually generally bring lower issues and stresses, than trying to fly solo from the beginning, but needing council help and support to achieve it. That gives others a vested interest, sometimes good, sometimes not.
You will remain priority housing post pregnancy because of overcrowding and being an under 18 mum with a requirement for education or employment.
When you ask for intervention, it doesn't always come in the shape asked for. Always remember that.
Approaching council services as a very young 'untested' mum, wanting things, is not generally a great position to be in. (though you can be lucky)
Once you are a few months in as a parent, have found your feet, and pregnancy hormones have settled down, you will be so much more equipped to deal with the range of professionals and their agendas, that seeking housing is going to bring. Your baby wont be 'abstract' to others, it will be very real and you will have bonded and learnt a great deal, including if there might be any additional needs.
You will have a child that you are a confident proven able mum to, and will slot better under 'family housing needs' rather than 'needs our CIN overview.'
If you start getting (wrongly) treated as the later, your protective instincts will be fuller formed and you will better identify and know when to back away or remove yourself and your child from things that aren't in your interests, regardless of what's being dangled.
The biggest piece of advice I can give you, is during this tricky balancing act, be concerned about your education and use the rights your generation have to it, and the responsibilities of others to ensure it, wisely. Your future self and your child/ren will thank you. It just wasn't possible when I was a very very young mum, and you can see the impact on my earlier children vs when I was able to get an education and better prospects, on my later children. It isn't age that's the big issue, it's lack of world vision and knowledge that come with it, you can do things to accelerate it.
Also it may seem odd to mention it now when you're at the stage you're at, but I want to put it in your head, because quite simply: always have a plan, or you end up just being a piece of someone else's plans.
With the right planning, work, and aptitude, it is nowadays possible to be a 19/20 yr old mum at uni with a child at nursery or in reception, and emerge a 22/23 yr old graduate mum with prospects. (Also possible to suspend and rejoin studies if needed) If you have a supportive deeply invested family, it's an easier journey.
Youth gives energy and it will carry you through if you use it wisely and there are benefits to being a very young parent that don't get much air, as people don't want to encourage it, but it isn't all negative, far from it.
Wishing you luck, strength and happiness whatever your decisions. Look after yourself and don't forget to enjoy it too, it passes so quickly.