OK, first here are a few of assumptions I've made which may or may not be true in your world:
1. The Elves' power has been on the wane for several centuries (or longer) and this decline is general knowledge.
2. This decline in power is due, at least in part, to the Elves' slow reproduction rates. Some also see a dilution of Elven blood (i.e. creating Half-Elves) as part of the problem.
3. The Elves (at lest the house of JK) believe population growth is the key to re-establishing Elven dominance AND they are working hard to keep their plans secret.
Depending on how evil the Elves are and how much control they have in the UK I think using Half-Elf women as incubators would be an option the powers that be would favor. First, if they rounded up a bunch of young, healthy Half-Elf women to be used for this purpose it would slow the rate of Half-Elf population growth. Second, it would keep Elven women from being exposed to the pain & hazards of childbirth (which are not insignificant in a pre-modern world). I'm thinking here of something like a 3rd level necromancy spell that transfers a fertilized egg from one willing woman to another willing woman. The recipient might only be willing because she faces torture or worse if she doesn't accept the transfer, but that's not really an issue if the people running the program are evil.
This also gives you some options as a DM to drop hints to the PCs about strange goings-on that they investigate. This leads to them finding more clues and eventually (maybe) put the pieces together. For instance, they might hear rumors that a decade ago riders from the house of JK descended on a few small villages and hauled off all the Half-Elf women of child bearing age. This was the start of the experiment and now that things are going well with the program, Half-Elf women are being recruited with incentives to their families if they come willingly. The party might also find a large, walled-in settlement where food and other necessities are brought in from the surrounding area. This would be the area where the children are being raised, commune-style so as to better instill in them the virtues of the Elven kingdom and the House of JK.
The clone spell is also a good option (and there could be two programs running simultaneously) if they have a way to get around the requirement that the donor die before the clone is activated. The other problem is that most people (I would assume) do not like the idea of their clone(s) running around in the world. Our uniqueness is something we value, so it seems unlikely to me that those in power would allow themselves to be cloned for the purpose of increasing the population. If they cloned the youth created from the incubator project discussed above, however, that's an easy way to increase numbers AND the youth would be much more accepting of it because they don't know it's not normal. Even if this modified spell is still 8th level, a few high-level casters could take a population of a few dozen kids and double it in a few months. A 15th level caster can only cast it once per day, but that's 30 new Elves every month.
Of course, it will be decades before any of these children are of reproductive age but that shouldn't be an issue for the Elves. If you're playing them as old-school with life spans measured in millennia (and if you're going to use this plot I highly recommend it), a few decades wouldn't seem like that long of a wait for them. If you've already established that Elves have the (shorter) 5e life spans, the PCs might learn that there is a small minority of Elves with pure blood whose lives span thousands of years. All the other "Elves" are impure, with human blood running through their veins. As long as there are at least 30 or so pure Elves, the incubator plan could work.
This type of situation leads to a whole bunch of issues which you and your players might not be comfortable with. There might be relatively new laws preventing the intermixing of the races, limits on the number of children non-Elves can have, etc. Sadly, there are all kind of real world examples you can draw from for inspiration. The one upside to this dystopian game setting is that the PCs get to take an active role in dismantling it (the appeal is something like Inglorious Basterds).
One other suggestion: the modified clone spell and the incubator spell are available to wizards and clerics (or maybe only clerics), at least Elven clerics. This means that the officers of the religion of state have an active role in the plan. The spells should probably also have a high value material component. Either a rare and costly item that must be imported or maybe some sentient being's vital force. The vital force might be drawn out of a "willing" being during the casting and age him like a ghost's effect. Either one of these options provide more opportunities for the PCs to pick up on clues they can piece together to figure out what's going on.
