They didn't start on the first exactly, it's just the most visible bit done so far. The club is working with Tim Lobb, who has rebuilt the bunkers on the third, the ninth and the thirteenth. The first was rebuilt because the green was not performing; it was a 20s green built by Hawtree and Taylor -- Colt's original was right on top of the hill -- and the green was very slightly bowled, which (a) meant it held water more than desired and therefore always had conditioning issues and (b) was out of character with the rest of the course. There was some discussion about whether to put it back on top, but the reason it was moved -- the severity of the climb -- is still true, plus there is now a very large water main up there which would make it basically impossible.
There are a number of issues around the course that need to be addressed. The seventh is odd and awkward, and the sixteenth has been changed entirely by tree planting up the left, designed to protect the houses on that side. It's very clear that Colt designed the hole to favour an approach from the left; but it's equally clear that isn't practical nowadays, so someone has to figure out another plan.
There is a possibility of a new back tee on 17 which would restore the strategy of the hump for everyone except the real bombers -- right now far too many ordinary golfers can just blow it straight past and ignore Colt's brilliant strategy. Tim and I found a spot for a new back tee on 14, right against the boundary fence, that would make it about 235 and return it to the extra-long one shotter that Colt designed, but I'm not sure there is much appetite to put it it. The fourteenth green was softened considerably in the 1920s; the front to back ridge used to be much more severe. Which is pretty hard to believe actually!
I think SGH is in many ways Colt's most interesting course. It gives the lie to any suggestion that he didn't build severe greens -- the tenth (which is also my favourite hole on the course) is still extreme, and a number were softened (by Colt) himself immediately after the course opened. George Duncan, who played in the opening day event, four putted one of the greens on the back nine, which caused a certain amount of stink. The eighth is obviously famous for the scale and drama of its bunkering (which I think will be put back at some point in the next few years) but it is also a pretty substantial piece of construction for 1913 -- you can see where he got the fill to build it in front of the ninth tee. Quite clever -- they excavated muck to create the little valley and used it to build the green, while at the same time giving visibility to the ninth drive (the eighth at Swinley is another hole that represented a substantial piece of construction for its time; next time you play there, look behind the green and you'll see the borrow pit where he got the muck).
The creation of the course is pretty well documented. Bernard Darwin visited the site at least a couple of times during construction and wrote about those visits.