Excellent cutting there! As far as a finish goes, any oil will do. I use a 50/50 mix of boiled linseed oil and mineral spirits. The linseed oil will impart a golden yellow color to the BB and bring out any grain in the wood. The mineral spirits thins the oil allowing it to flow around the wood and cutouts easier and will allow the oil to dry in about 24-48 hours. If you don't want to make a mix, Danish oil or tung oil will work and do just the same. I use the 50/50 mixture because in the long run, it is cheaper.
I use an old baking pan to soak the project in. If the project is bigger than the pan, you can brush it on. Just make sure you cover the wood evenly. Use rubber gloves to do this. After the wood is coated, wipe off any residual oil with a clean rag then set aside to dry.
Once dry, I use a semi gloss spray lacquer as a top coat.
One bit of warning. Do not lay oily rags in a pile, they can self combust! I hang them over the edge of a metal trash can for twenty four hours, then you can toss them.
I store the oil in an old quart mason jar.
I use this method for every project I cut with great success!