What is the modern bottom bracket compatible with this bicycle? It's 35mm diameter and it's a French threaded BB.
Having a quick look at the offerings of BBs, I can see most of them have a square taper axle while my original crankset is different with a round profile/interface with little cut(s) for cone bolts. Does this mean I'll need new crankset as well?
You can get cartridge square taper French BBs from various companies like Velo Orange, IRD, and SKF. The VO ones are modest priced, and so most of the time the practical solution is choose a new square taper crank and get the VO French BB in the needed length. Note that the chainline needs of older bikes is often further in than whatever the on-paper matching spindle length will lead you to on current crank specs, so if you're keeping it 5/6 speed freewheel in back, doing it right starts with ascertaining what chainline you want and then doing the spindle math. Another option: Phil has made 24mm external cups in French thread if you wanted to run a modern road crank for example. Finally, NOS or used French cup sets are not rare and can be found on eBay etc.
You have cottered cranks. If the cups that came with them are toast and you wanted to keep using them, that would mean getting an NOS or used cup set as above. If your spindle is in reusable condition, it will probably work with whatever such cup set you come up with, but there may be some luck of the draw and/or research if you wanted higher certainty on that point. All that would usually matter there is whether the hole in the cup is big enough for the spindle to pass through. If the spindle was not reusable, you would need to source a new one of those as well. Once you had a set of compatible parts, you'd then often need to get appropriate new cotter pins (usually old ones are not in reusable condition, but there are exceptions, and care taken in extracting them is part of that), and once you have them there is frequently a need to manually profile them correctly for the cranks at hand. Usually none of this is worth it because cottered crank chainrings can't practically be replaced and were usually way too big in this era even if they're still in okay condition.