Yes the building warrant is what allows you to start building and is what you are more familiar with in terms of building control. In terms of timescales it’s at least 8 weeks for planning and then circa 12 weeks for the building warrant with a gap between them to prepare the building warrant drawings although you can start some of the drawings before you have planning approval with the obvious risks. In my case it took 12 weeks for planning approval, 8 weeks to get the warrant drawings (the structural engineering drawings took weeks to get) and then 8 weeks to get the warrant approval.
I’ve founding dealing with our local council BCO to be a positive experience. He’s very pragmatic, genuinely wants to see you get the house built, easy to get a hold of, and quick to respond. If you’re using an architect they’ll typically do all this for you. The councils also use the Scotland eDevelopment platform make the various applications and upload the required documents.