Amazing chairs! I didn't appreciate that they were quite so magnificent. There's no question but you have to try and remake those in the original timbers. Substituting a close visual match is all well and good, but for a genuine "legacy project" like this you need the real deal. Unfortunately Laburnum can be quite distinctive, you'll often find faint traces of medullary rays on the quarter sawn faces, not as pronounced as Oak but it's there none the less, and matching that would be a real challenge. I think you'd get fairly close with Sonokoling Rosewood, but a knowledgeable person scrutinising the work at close quarters would spot the subterfuge and regard it as just plain
wrong!
Incidentally, one fascinating snippet for you. Most people know that bows were often made of Yew, but what isn't so commonly known is that Yew was only one of the timbers commonly used for bow making, indeed there's a Plantagenet statute that lists the timbers to be used for bow making. Yew, Hazel, Ash, and a mysterious timber called "Hoburn". There's a strong chance that "Hoburn" is actually Laburnum, and a Bowyer I met had actually made a bow from Laburnum and regarded it as every bit as suitable as Yew or Ash. The doubt comes in because the first recorded example of Laburnum grown in Britain only came later, but personally I don't find that a valid objection. It's a common enough European tree, so for all we know the Romans could have brought over and planted a few examples!
One way or another I'm fairly confident you'll get your Laburnum, it may take a bit of time to track down some knot free, straight grained stuff (these "park grown", ornamental trees, tend towards sprawl, so you rarely get the long clean bole that furniture makers want) but with patience I've no doubt you'll turn something up.
The bigger challenge is the Elm. And it's a challenge that has at least three components.
The first part of the challenge is English Elm versus Wych Elm. Marcros has already pointed out that you may have better luck sourcing Elm in Scotland. He's absolutely right, but it's almost certainly going to be Wych Elm. For example a few years ago there were quite a few mighty Wych Elms felled in the grounds of Edinburgh's Holyrood Palace, I got a few boards and for a little while had a decent business using it for furniture that went to clients of Scottish ancestry in Canada and America. Here's one of those boards, and very nice it was too! The good news is that boards from that same source are still in circulation and still available in a few yards.
In the general run of things I'd judge Wych Elm as a reasonable substitute for English Elm. But if your chairs are genuinely significant then you may conclude that "close" isn't actually close enough! In that case you're left looking for the far rarer English Elm.
The second challenge you have is one of dimension. Elm tended to be a fairly whopping tree, so really wide boards capable of making an unjointed seat constituted a decent percentage of all the boards out there. But the bigger problem is likely to be thickness. Specimen windsor chairs that I've measured often had particularly deep saddling to their seats, no surprise as even an extra 1/4" in thickness adds masses to the visual impact. Consequently you'll need a 2 1/4" or 2 1/2" thick board. Unfortunately most mills will cut quite a bit thinner. It's heart breaking to finally track down an English Elm board and discover it's been converted at 25mm, 35mm, or even at 50mm.
The third challenge is finding two Elm blanks that have a harmonious and shared grain pattern, such that they sit comfortably in each others' company. This probably means either finding consecutive boards from the log, or a single board large enough to yield two seat blanks. If you go back to the photo above you can see that it's actually marked up for windsor chair seat blanks and that each seat will have a similar catspaw pattern of pippy grain towards the centre. With two consecutive boards this yielded six chairs that formed a unique and harmonious set for a breakfast table.
But given that you're trying to match chairs with extant specimens you may prefer a more neutral grain pattern, such as this English Elm Windsor seat blank,
The right boards do exist, but they're extremely rare and it will take a lot of patience and detective work to find them. I've got a couple of boards of 65mm English Elm, actually I've hung on to them in the hope of a commission for a matching pair of Gothic windsor two seater settees, so not that distant from your project! I've carried a picture in my minds eye for many years of them sitting opposite each other, the focal point of the entrance hall to a country house or as a counterpoint in an uber cool city apartment.
The point is that once or twice over the years I've had the opportunity to replace these boards, so even though they're rare they are not quite in the unicorn droppings category!
Again, a great place to start would be a post on the WoodLots forum. With luck you'll find a tree surgeon who's about to remove a big English Elm, so you can specify conversion to 65mm or 70mm thick boards. You'd still have to wait three or four years for air drying, but that would provide bags of time to source some truly exceptionally Laburnum. As I always say, cabinet making suits the tortoise more than the hare as it's all about playing the long game!
Good luck!
