Shared ownership doesn't need to be brand new, I've seen older SO properties on rightmove, I usually notice them because I think 'wow that's cheap' but then see it's only for a quarter of the house.
I'd recommend looking at the Moneysavingexpert guide to ISAs because I think you can 'reserve' one while you still qualify for it, and then just stick a few quid in and top it up later, so worth getting if you qualify (I don't so don't understand them properly, but I know they're almost always a no brainer for those who qualify).
www.moneysavingexpert.com/savings/help-to-buy-ISA/
Have you talked to a broker or at least looked at some online calculators to see if you qualify for the amount of mortgage you need to buy the house you need? Then you can work out what sort of deposit you might need.
Eg, if you qualify for a £150k mortgage, and you can get a 4 bed house for £180k, you need to save about £30-35k for a deposit and fees.
However, if a 4 bed house will cost £350k, you would need £200k+ as a deposit, which doesn't sound doable, especially as you need to pay your rent and feed/clothe etc your family, including presumably at least 3 DC as you are looking for 4 bedrooms. If a 4 bed house is going to be beyond your reach, tehn you need to be looking at cheaper areas, a smaller property, or shared ownership.
LA = local authority, ex council house, which are usually cheaper than those that have always been privately owned, and are often a good size, well designed with bigger gardens. Often in perfectly acceptable areas too. I live in one myself.
The MSE mortgages guide will probably also be helpful
www.moneysavingexpert.com/mortgages/mortgage-guide/