Pictures of your bath as it is now will net the best suggestions.

Generally, existing trim gets removed then reapplied on top of the b&b.

Since the actual bathroom isn't shown, I'd have the installer draw every intersection on the wall before cutting anything. In general, don't ease each batten into the existing trim. Remove the window casing and baseboard, install the panel and battens, then reinstall or replace the trim on top so the edges look intentional.

At the base, the battens can die cleanly into a flat baseboard if the baseboard is at least as thick. If the colonial profile is thinner or has a rounded top, I'd replace it in that area with a simple flat board and small cap. Around the medicine cabinets, stop the battens at a consistent gap from the cabinet trim rather than adding tiny filler strips.

For the tub splash area, use PVC or another moisture-safe panel and battens, not MDF. Painted drywall between battens is water-resistant at best, not waterproof. Keep the radiator's required clearances and make that section removable for service instead of boxing it in.

Here is the bathroom as it is now.