Being in the floor covering game for 37 years all I can say is the timber does not do well with a lot of moister and before no time the timber will expand to a point where the boards will fall off. Have you looked at the new ceramic tiles ghat look like timber ?

The gaps shown look no more than 5mm thick mostly, so I would address the floor, as preparing this will be the only way to get a worthwhile result. A self leveling compound poured over the slab will rectify the surface and fill part of this gap, with the flooring p,plank doing the rest. Or opt to have a foam underlay put down prior to the board to also provide an additional 2mm. You can always install quad around the perimeter to conceal this gap, which most fooring installers allow for flex within the boards regardless. Temperature change will mean they will expand and contract, and this tolerance is Important so they don't jar and pop out. Some gap is actually required. I would go with the first option to get the best result.

We did floating oak floors, looks great and have just realised needs more rugs for winter. So the boys plus a large rug could work for all concerned. Noise is hard to relate to without 3 boys running around here. You can get acoustic underlay but for the extra I would get rugs in the critical flow areas if you prefer the look of timber over carpet. Lastly, carpet is relatively cheap and helps to eliminate noise and most likely replaced when they are older/leave home. Then you can have timber floors, or make other choices to suit.

some of the older carpenters and cabinetmakers will have the necessary traditional planes for cutting these profiles or you could have one made out of plaster, cement.or builders bog moulded out of a cast made from the remnants of the original board or try a high tech solution with a copy made with a 3D printer or two separate lengths of moulded timber attached together to approximately match the original but you'll need to research plaster works, restoration companies, specialist carpenters and tech colleges for a computer, woodworking or art student keen to try an interesting project but you may find that it's quicker and easier to replace the whole section with an unmatched section of plain skirting board or try stealing a section from somewhere else in the house where the skirting board is hidden behind a bed or lounge and replace the hidden area with the plain board...best of luck with your repairs