To find out the BB size, remove the crankarms and measure the width of the BB spindle. A dial caliper is the best tool for this.
You do not neccessarily need a new wheel. You can use spacers to widen the width of the wheel. If you use a quick release, you will also need a longer axle.
Whether the crankset will work depends on the chainline of the new frame. This is dependant on the BB spindle length on the drive side and the position of the chainrings relative to the end of the spindle. This is something you need to discuss with your framebuilder.