Here's a good, brief summary of the matter of bottom treatment during the period in question, from The Line of Battle: The Sailing Warship, 1650-1840. It's the relevant volume in the series Conway's History of the Ship, which, generally speaking, is an excellent place to look for the current state of scholarly thinking on such topics. The author of this particular chapter is Brian Lavery, of the National Maritime Museum at Greenwich. The quote is from pp. 143-144.
"The underwater hull of a ship was afffected by two main problems. Weeds and barnacles could attache themselves to the planks, and drastically reduce the speed of the ship. Even more serious was toredo navalis, or ship worm. This flourished in the tropics, and could eat its way along the planks of the hull, making the ship totally unsafe.
"Until the late eighteenth century, the two problems were generally tackled separatelyl. For protection against the worm, ships on the way to the West Indies or India were sheatehed. A noxious composition, usually a mixture of hair and tar, was put over the underwater hull. This was covered with light planks, usually 1/4in thick on British ships, but thicker on French. It was hoped that the worm would confine iteself to the sheathing planks, and not attempt to penetrate the tar.
"For protection against weed, other antifouling compositions were used, and put on top of the sheathing if that had been applied. In the seventeenth century the most common was a mixture of train oil, rosin or sulphur. This gave a white appearance, and so was known in Britatin as 'white stuff.' Around the middle of the century some ships were tallowed as well, though this was probably a way of getting a smooth, frictionless surface, rather than forthe protection of the hull. A cheaper alternative to white stuff was 'black stuff,' made from pitch and tar. Though the work of modelmakers and marine artists suggests that white stuff remained prevalent, most British ships of the first half of the eighteenth century were in fact coated with black stuff. In the 1740s, 'brown stuff' became standard for British ships; it was black stuff mixed with sulphur, and was said to have some of the advantages of both compounds."
It sounds to me like "white stuff" would, when fresh, be a yellowish, greyish, off-white color. The Model Shipways and Testor Acryl ranges of hobby paints both include colors called "hull tallow," which look about right to my eye. After a hull treated with white stuff had been in the water for some time - well, I'm not sure I want to know exactly what it would have looked like.
The next few paragraphs of the book deal with the introduction of copper sheathing. Mr. Lavery says the first experiments with it took place in the late 1750s. The first attempts got off on the wrong foot, because the shipwrights were nailing the copper sheets to the hull planking with iron nails. Copper plus iron plus saltwater equals electrolysis; within a year or two the copper around the nails got eaten away, and the copper started to fall off. (The iron nailheads holding the planking to the frames also posed problems if they came into contact with the copper.) The solution to the problem was to make the relevant bolts out of copper instead of iron. Here's the essential paragraph from Mr. Lavery's chapter:
"In 1775 the struggle for American independence became open war, and in the following year it was decided to copper some small ships, despite the risks. In 1779, afer the war had become general and the British navy was outnumbered, it was decided to copper ships of the line as a matter of policy. There was not time to drive out the old iron bolts, so efforts were made to keep the copper from the iron by covering the hull with tar and brown paper before coppering. This was enough as a temporary measure, but aftetr the war ended in 1783 it was decided to r-fasten thewhole fleet as ships cae into dock for repair. The French, who had never been far beehdind int he development of coppering, adopted it at the same time, adn it became universal in other navies."
There was one more step in the evolutionary process - after the period covered by that particular book. In the mid-nineteenth century it was discovered that a more durable sheathing material could be made by adding other elements to the copper. "Muntz metal," sometimes known as "yellow metal," was introduced in 1830; initially it was 50% copper and 50% zinc. In 1846 Mr. Muntz changed the formula to 60% copper, 40% zinc, and sometimes added a small percentage of tin. (My source on this is George Campbell's China Tea Clippers.) For the next couple of decades, ships were described as being "yellow mettalled," "coppered," or "brass bottomed." (Brass, according to the dictionary I have in front of me, is typically 67% copper and 33% zinc.) Sometime in the mid-nineteenth century, "anti-fouling paint," which had particles of copper and other metals mixed in with a vehicle in brushing consistency, came into use. I don't have any precise dates, but my impression is that it was initially used for small craft and eventually, of course, came into use in large vessels. By that time the wood hull was giving way to the iron one, which made metal sheathing irrelevant.
Bottom line: Le Soleil Royal's bottom would have been painted with either "white stuff" or "black stuff" - probably the former. The Victory and the Constitution would have been copper sheathed, and we know for a fact that the Cutty Sark was sheathed with Muntz metal.