Question: What would be the best route to get these speedlights to work?
My recommendation would be to return everything and start afresh. With a Godox speedlight and transmitter (e.g., TT685N speedlights and X1T-N or upcoming XPro-N transmitter). The expansion options and cross-platform support are much much better with Godox than with Yongnuo, and you can mix their manual-only and TTL gear without any issue.
If you want to keep the YN-560-TX, then you must give up the idea of having TTL or HSS. This is a manual-only radio transmitter. You will get remote power/zoom control. And for convenience/cost, I'd recommend going with YN-560III, YN-560IV, YN-660, or YN-860Li speedlights, since they all have RF triggers built-in, so you don't have to remember to add them on (as you would with the YN-568EXII flashes).
If you want to keep the YN-568EXIIs, then return the YN-560-TX, and get a YN-622-TX transmitter and two YN-622N transceivers for the flashes. This will have a big advantage over the YN-560 system in giving you TTL and HSS, but will have the inconvenience of more items of gear you have to remember, slot together, and batteries to carry/recharge. The YN-685 has a built-in 622 receiver, but no optical slave capabilities, or RF master capability.
I wasn't aware that Yongnuo had come out with a Nikon version of the YN-568EXII (only the YN-568EX), so I would caution you to check that it's the Nikon versions you have. The listing you linked to is actually for a Canon flash. If you plan on using these on the hotshoe of your camera or of YN-622 triggers, it will be manual-only, with no TTL/HSS function, since it has the five pins for Canon on the foot, not Nikon's four-pin layout.
If they are Nikon versions, they can probably be CLS master/slaves in Nikon's "smart" optical system, so you could use that instead of adding on radio triggers. Your camera body might have a master/commander for this system in its pop-up, if it's a prosumer body (i.e., D7x00 body or above).
They're compatible. Kind of.
You can use them as manual only flashes with the Yongnuo YN560-TX and the receivers listed in the following paragraph. Or you can use them as full TTL flashes via the Nikon CLS/AWL optical wireless protocol. But the physical connections on the feet of the YN568EX II are not hardware compatible with the Nikon TTL system.
You just need a radio receiver for the two flashes that uses the RF-603/RF-603 II/RF-605/YN-560-TX/YN-560 IV/YN-660 protocol. Of course the YN560-TX is a manual only controller and won't allow you to use the TTL, HSS, etc. capabilities of the YN-568EX II. With proper receivers (the pin patterns on the hot shoes of the receivers need to match the pin patterns on the feet of your flashes - so you need Canon compatible receivers) it will allow you to control manual power and zoom head position from the YN560-TX. Note that the only difference between the Canon and Nikon versions of the YN560-TX is the flash wakeup feature (The Canon version can't 'wake up' Nikon flashes and vice versa). This works because the only things the YN560-TX gets from a camera body (Nikon or Canon) are the "fire" signals from the center pin and the "wakeup" signal from either a Nikon or Canon camera.
The YN-568EX II has a built in optical receiver that can use both the Canon and Nikon optical communication systems. Any Nikon i-TTL commander (e.g. SB700 flash, SU800 controller, or the pop-up flash in most recent Nikon bodies except the D3xxx and D5xxx series) can control the YN-568EX II wirelessly. Of course going optical comes with range, line of sight, and multiple photographer disadvantages compared to wireless radio control. But the YN568EX II has Canon connections on the physical feet of the flash. Yongnuo never got around to making a Nikon version of the YN568EX II that has Nikon physical connections on the feet of the flash.
The YN568EX II can also be used as a simple slave flash. As a simple slave it is not receiving any kind of coded communication, via either radio or optical pulses, from the camera or camera mounted flash. When set to 'S1' the YN568EX II will fire when it senses a bright flash. Any bright flash will fire it, whether from your camera, another camera, or off camera. When set to 'S2' it will (hopefully) pause for a pre-flash fired by a TTL flash and (hopefully) fire in response to the main flash pulse from the other flash.