As usual, I'm with Tony on most all things 'Automation' - I have a different background than he does (I'm a full-time automation engineer, but I work almost exclusively with custom machine projects in the automated assembly industry) and I also happen to own a small shop of my own.
I think the first thing to know, which is absolutely critical, is that you're going to either need to hire someone who knows automation, contract it out, or commit to sending someone for robot training, and probably PLC training (at an absolute minimum.) Tony downplays just how unique he is, and how talented, but the number of people out there who can design and integrate a robotic automation cell for a CNC machine, and do it well, is extremely limited. Lots of folks can do a piece, or two, or the big piece, but finding an idvidual who has a great understanding of both the Mechanical and Electrical Engineering aspects, as well as fixture design, robot programming, EOAT design, and PLC programming (and doing it all in a way that is SAFE)....well, you're better off chasing the gold at the end of the rainbow.
If you think it's something you can just buy the pieces and figure it out - well, you'll wind up looking like a YouTube machinist who has no clue what they're doing. Except the CNC programming part of these projects is definitely not the hardest part. And there are a lot more parts.
And if you're going to invest half a million dollars or so on automation equipment, then think you'll save money by trying to integrate it in house, I'd suggest not doing it at all. I just visited a customer in Buffalo who is attempting this on a palletizer line (it doesn't get much easier) and is now paying our guys to fix all the mistakes they made, which pretty much involves going back to the start. It would have been significantly cheaper for them to buy the palletizer cell from us.
Have I sufficiently scared you yet, while proving what a superior d-bag I am?
Seriously, I love seeing shops get into automation. It's what I do, after all. But you're asking the question a TON of shops are asking right now: how do we automate our job shop? It's a great question, and I'm not sure anybody in the industry has a good answer yet. I've been looking at it for the last 2 years, both as part of my full-time job, and how I would automate my own shop, and have yet to find a great answer.
Honestly, with all the 'built-in' automation capabilities already in a lot of newer lathes, you'll be hard pressed to add much with a robot, especially considering you'll have to integrate that, versus buying a bar feeder, tool changer, part catcher ,sub-spindle, live-tooling, etc. that all comes in the same box.
If you want to see cool ways to use that UR on equipment that doesn't have any of those features, check out Mr. Rudolph's Instagram. Instead of buying servo doors, wiring into the controller, etc. you just make the robot do it all. Hence the appeal of URs.
I would seriously recommend having an automation company come in and do an 'automatability' study. I bet if you reached out to MotionGuru (who has experience in this field and is in the Pacific NorthWest) he could either recommend someone, or give you an idea of a price to evaluate the project.
This is normally where I pitch myself for this type of work, but you're on the opposite coast and I'm busier than I can handle right now. But if you want to reach out via email for a sanity check, I'm happy to help out here and there as I am able.