Sure Start grant & UK benefits

I’m currently 37 weeks pregnant and I’ve used some online calculators which have determined I’d be entitled to UC and other benefits once my mat pay drops to SMP after 6 weeks Am I correct that I can only start claiming benefits and making an application for the grant when I’m on SMP, so after the 6 weeks? I don’t mind waiting but I’ve seen that it can take up to 8 weeks from submitting the paperwork, having it reviewed ect to then receive payment so I was just wondering whether there’s anything I can do now to get that process started? I’m not receiving any benefits currently but things will be really tough when I’m just relying on SMP so any advice would be really appreciated ❤️