LEAD, will ONLY be in PAINT. The only other place that lead was prevalent was in motor fuels and that went away in the early 1970's.
It will NOT be in Varnish, Stain, Shellac or any other clean finish.
That is why it is referred to as LEAD PAINT. It comes from the LEAD OXIDE used in the colorants to make the paint white.
The only way it would pose an issue to your kids, is if they were to gnaw on the PAINTED structures and chew down to bare wood.
Or, if they were to create PAINT CHIPS and then EAT them.
As long as the paint is solid, and there is no cracking, peeling or chipping, there is no danger to you or your children.
If you choose to sand any painted surface, just wipe it down with damp rags and then dispose of them. DO NOT LET YOUR CHILDREN SUCK ON THE RAGS.
Also, vacuum up any dust and then dispose of the bag if your vacuum uses one.
If your are at all concerned, buy a LEAD TEST KIT and test your painted surfaces.
As for painting your stained and varnished trim white, go ahead and scuff sand them, put a good primer on them and then paint them with a good quality paint.
But, to allay any of your fears, unless your painted surfaces are in such a poor state that they are peeling, cracking or chipping, or if your children suffer from a medical condition called "PICA" I wouldn't worry about LEAD PAINT.