The baby isn't here yet. So the money you plan to spend taking care of it, put that money in a bank account and soon you'll have enough for a rental deposit. Your boyfriend pays rent, council tax, utility bills as it is. So all that needs to change is he starts paying those things on a rental for you two instead of his mum's place.
Do you both have a job? Private rentals will want to check your income before they'll rent to you. Without sufficient income they'll want a guarantor. His mum will probably refuse since she's seemingly pretty nasty. She also may not have enough income herself to be accepted as a guarantor. If you don't know, a guarantor is a person who legally agrees to pay the rent if you don't pay it for whatever reason.
If you don't have sufficient income or a guarantor, you can offer the landlord 6 months or a year's rent upfront. This helps mitigate their risks in renting to you. They may or may not let you pay monthly after the initial rental term ends. So it's wise if you spend the time saving up the next 6 months or a year's rent in case you have to pay that way again. The landlord can't legally ask for huge amounts of rent in advance like this, you have to offer it and they don't have to accept.
To rent or buy, visit an estate agent. Most have a lettings (rental) section as well as properties for sale. They can tell you all about the various fees for property purchases.
You'll need a mortgage from your bank (or a bank) so you can visit the branch and ask to speak to someone about that so you can find out how it works and how much they'd lend you and how much deposit you'd need (they don't usually lend 100% of the property's value).
It should be free to speak to estate agents and bank staff. If you purchase a property a solicitor will be needed too, speaking to them isn't free, it's expensive.
In very simple terms, to rent, you see what is available at the letting agent, arrange with them to view it, tell them you want it, they complete checks and if you pass these, you pay the fees and the deposit and the advance rent (one month in advance is usual), then move in. Most rentals are available almost immediately and at the end of your tenancy if you're asked to move out you don't get much notice of that either.
Buying takes a lot longer because everyone has to have sold whatever they currently live in (not you because you don't own anything yet) and also to have found somewhere to move to, so you end up with a chain of people who are all depending on each other for the sales of all the properties to go through without a hitch. In simple terms, to buy, you look at for sale ads in estate agents and arrange a viewing of any you're interested in, once you've picked one you arrange searches. These tell you things like eg if the house was built on a massive sink hole and the previous three houses built there all lasted ten years before being sucked into the hole - and other such disastrous things you'd like to know about before purchasing. (I'm exaggerating by the way, I'm not aware there are any sink holes). Then it's a survey to give a basic overview of the state of the building eg noting if there's a hole in the roof etc. Which might affect the price you're willing to pay for the property. Then solicitors draw up the paperwork, both parties exchange contracts and on completion day you pay the purchase price/collect your sale price and collect the new keys/give the old keys and move out/in. Everyone in the chain on the same day.
Note that a surveyor and a solicitor works for you and should have only their own your interests at heart. Estate agents and letting agents work for the property owner and have their own the property owners interests at heart. They're sales people on commission. They're not going to point out all the flaws to you.