(written from a Production point of view)
First appearing as prototype kit-bash studio model in the debris field of Star Trek: The Next Generation's episode "The Best of Both Worlds, Part II", the design of the Template:ShipClass was upgraded to a full worthy studio model, to represent a major class of starship in "The Wounded". As Michael Okuda explained on Doug Drexler's blog, "The original concept for the Nebula-class ship was to develop a design that was in the style that Andy Probert had so brilliantly established for the Enterprise-D. (You may recall that every other Federation starship in early TNG episodes was made with recycled movie ships.) Our initial hope was that Greg could use the same molds from the 4' Enterprise-D, but that he could add a bigger bridge and give it bigger windows. The idea was to suggest that this ship was a contemporary of the E-D, but it was a smaller vessel." 
The BoBW kit-bash models
Ed Miarecki designed and built two Nebula-class vessels to appear in the debris field of the "The Best of Both Worlds"-episode. Okuda remembered, "Nebula-class USS Melbourne. The original version of the Nebula. Ed made us two nearly-identical Melbourne models. I destroyed one of them to create a battle-damaged ship for BOBW2. (I recall being up in the art department one evening, Dremel tool in hand, cutting up the ship, while Patrick Stewart, in Borg costume, came upstairs to use the Xerox machine. He asked me what I was doing. I showed him the model and said something like, "See what you did!") Rick Sternbach helped with the battle damage, too. And Greg Jein contributed quite a bit of wreckage as well. The other Nebula model ended up on one of the side tables in Sisko's office in DS9. I think it was labeled "Melbourne," too."
The models, built from parts of Template:ShipClass AMT Star Trek model kits, No's 6618 and 6619, differed from its later definitive appearance in that they sported two smaller warp nacelles where the sensor pod was to be positioned and that the secondary hull was more elongated. Sisko's desk top model (having made two early appearances as desktop model in TNG: "Future Imperfect", "The Wounded"), representing the destroyed Melbourne was refitted with a sensor pod after its first few appearances in Star Trek: Deep Space Nine.
For "Emissary", the producers decided to use the more detailed Template:ShipClass studio model to represent the USS Melbourne, thus robbing the proto-Nebula from its name and registry. Stock footage of the now nameless distressed model is present in the episode (seen in the window of Benjamin Sisko's escape pod, just as it leaves the doomed USS Saratoga), making it the only studio model, besides the Borg cube model, to appear in both depictions of the Battle of Wolf 359 and its aftermath.
The physical studio model
The design upgrades for the Nebula-class were done by Mike Okuda and Rick Sternbach and the model was built by Greg Jein. In Drexler's blog, Okada elaborated further: "As so often happens with this kind of project, we didn't give Greg enough time to accomplish this, so we decided to retain the original scale of the Galaxy-class ship saucer. I suggested the original 'AWACS' pod in response to a producer's observation that the ship might otherwise appear unbalanced. Unfortunately, the AWACS pod didn't look as elegant as we had hoped in "The Wounded". Rick Sternbach came to the rescue with the cool triangular pod that we used in later episodes." The time constraints could also explain why the USS Phoenix is not equipped with reaction control thruster assemblies. Fortunately, Jein was able to save time by using the molds of the recently constructed 4-foot Template:ShipClass to cast the saucer section and the nacelles.
As to the apparent lack of impulse thruster on the model Sternbach commented in the same blog, "One of the rationalizations I imagined ages ago for having no obvious impulse grilles involved capturing and compressing the impulse fusion reaction exhaust and later releasing it from special non-propulsive ports. If the Nebula-class was employed in a stealthy surveillance mode, it would be smart to minimize all overboard emissions. Since most all 'modern' impulse engines involve little or no pure rocket thrust, but more of a sub-warp drive, one could say that the familiar orange Starfleet glowy exhausts could be modified or eliminated."
The model was used extensively, redressed (mostly done by Drexler and Okuda) several times to represent multiple vessels in later seasons of Star Trek: The Next Generation and the first three seasons of Deep Space Nine. 
When the shape of the sensor pod was reconsidered, Doug Drexler was also asked to come up with an alternative, "At one point, Dan Curry called Mike indicating that we may need an AWAC pod variant. We obliged by kludging one together. We ultimately did not use it.", Drexler remembered, who used amongst others, parts of a Miranda class model kit and computer keyboards. Instead the producers followed Sternbach's re-design. Drexler's part, measuring 12"x11"x6", turned up on 8 August 2010 as Lot #76 in the Propworx' "Star Trek Prop and Costume Auction", estimated at US$600-$800, where it sold for US$350 (including buyer's premium).
In 1994 the model underwent extensive refurbishment done at the Industrial Light & Magic workshop supervised by modeler John Goodson for Star Trek Generations. The model was totally stripped from its original paint scheme (white with duck-egg blue highlights) and a new more detailed paint scheme, necessary to meet big screen requirements, was applied (blue-gray with metallic blue highlights) to represent the USS Farragut (Cinefex, No.61, 1994, page 69). In its new livery the model was used to represent the USS Leeds, for the upgraded opening title sequence of the last four seasons of DS9. Since no dorsal shots were made for this sequence, the Farragut decals on top of the saucer were not replaced. The last time the model was shot, was for a battle scene in Star Trek: First Contact as an unnamed vessel (although the model was still wearing the Farragut/Leeds decals, the shot was composed in a way that name and registry were not discernible).
The physical studio model itself, worn and showing its age has escaped the 40 Years of Star Trek: The Collection and It's A Wrap! sale and auctions, and is still in the possession of Paramount Television, having been on tour as late as 2011. After closure of the Star Trek The Exhibition tour at the Kennedy Space Center, Florida, on 31 October 2011, the model was pulled from the tour circuit and remanded in temporary custody of Alec Peters of Propworx, Inc..  Damaged from years of intensive handling on the tour circuit, Peters stated in private correspondence with Ex Astris Scientia that, "It will be restored and then who knows where it will wind up, but won't be sold. It is part of the CBS archive now! I have already contacted Greg Jein about restoring it."
The CGI model
For appearances in the later seasons of Deep Space Nine and Voyager a CGI model was built at Foundation Imaging in 1997 by Robert Bonchune, making its debut in VOY: "Message in a Bottle". In a ploy that was reminiscent of what Jein had done, constructing the physical model, Bonchune said, "I got to have as reference, sitting at the front of my desk, the actual filming miniature! It was a good 4 feet across. In the effort to be expedient, I did "kitbash" existing parts of the CG USS Enterprise-D, but it still needed completely new parts and major changes to existing maps to match the miniature,". (Sci-Fi & Fantasy Models, issue 32, 1998, p.54) The utilization of the parts of the Galaxy model went a long way explaining why the CGI Nebula differed slightly in appearance from its physical counterpart in that both the center line section of the secondary hull and the saucer rim were slimmer, suggesting, on screen at least, one deck less each. The most significant difference was the change of the shape of the deflector dish housing to an oval shape akin to the Galaxy-class. Also, the CGI Enterprise-D sported the original gray-blue/duck-egg blue paint scheme, mapped onto that model from high resolution pictures taken from her four-foot physical counterpart, which was also transferred onto the Nebula model instead of the new paint scheme applied at ILM.