Sorry for the thread necromancy...
I am using this map currently working on an 'Amalgam Universe' map, combining various franchises, What started out years ago as me wondering how to combine SW/ST in one galaxy, to me recently wondering about adding a 3rd (Mass Effect), has now turned into a mapping/Lore project that at this time includes well over 300 franchises, the basis of all of which is Star trek and the Federation. And here's the problem... ST writers/designers/consultants are all on crack. LOL (and at the end of the day, they're still better than most)
The one thing I have always loved about ST was its more realistic approach, but unfortunately, writers 'science chops' and the shows various 'science consultants' (the guys who got D's in college?) have never really done a good job, other than figuring out all you had to do was 'reverse the polarity' of something to undo it. I have been making maps for 20+ years, some professionally, usually for RPG franchises, and a few for novels. I've never tried star maps before, but since my maps are widely acclaimed as "THE most accurate" by some folks, and how much I love Star Trek and scify in general, I figure I'd give it a go just for S&G. Its turned into quite the monster, but since I am retired now, its a great way to pass the time, writing 'conjoined lore' and making the map. But I was dumbfounded that there are no good maps of ST - most do not agree with each other. I have found a few that are official in one capacity or another that are close enough that I can at least tell its the same areas being mapped (for example, maps often swap the Romulan & Klingon empires around). As part of my process, I use tons of real-world references, whenever I can (and have pissed off at least one best-selling author when I explained to him his favorite region to write in was 'impossible'). Magic is one thing, pretending physics don't even exist is something else entirely. So applying this to Star Trek, we have a solid, CONFIRMABLE distance of at least place - Vulcan. Vulcan is located around an actual real star system in the Milky way - 40 Eridani. Its 16.5 LY from Earth. Applying this to the few 'constant' maps we have, the Romulan neutral zone is between 32-35 LY away from Earth. Yeah, I was surprised myself. All of Trek takes place in only a tiny place in the Milky Way, and the few galactic maps that show otherwise are DEAD WRONG. I've tried to extrapolate outward (resize the maps) to match some of those Galactic ones, and Vulcan winds up at least a 1000 LY away, which is ridiculous. If that were the case - and we've seen ships get to Vulcan in just a few hours - then the 75 year journey of Voyager, being 75K LY away from Federation space, should have taken under three months. Believe it or not, Voyager actually handled the distances involved the most realistically. Everything else travels at Game-of-Throne speeds (wormholes!)
So the answer to the original question is 32-35 LY, depending on which maps you go by, and the fuzziness of borders. At Warp 9+, you 're traveling about 80 LY per month (going by Voyager's estimates), and it should take around 12 days at maximum warp to reach the Romulan border. Picard would have NEVER made it in time for the battle.