You are correct, under Schedule 1, you could apply to the court to allow you to remain in the property until your youngest is (usually), 18. Whether you will succeed however depends on your ex's ability to manage without his equity, and to pay the mortgage in addition to his own accommodation.
What would you be able to contribute in terms of outgoings? As the mortgage is in his name alone, the problem is that if he doesn't pay it, the house will be repossessed and to be honest, I think the only way you will succeed, given the extremely long term involved, as your youngest isn't even born yet, is if your partner is very wealthy and can be shown to be able to afford to wait 18 years to receive his equity and at the same time pay the mortgage.
Schedule 1s tend to be successful where there's been a very lengthy relationship, the children are over about 11 or 12, and the dad (it usually is the dad) can easily afford it.
The difficulty is that you weren't married. If you were, it would be a different ball game altogether, but I'm afraid you may face an uphill struggle.