Vulcan annular warp drive is an application of Cochrane's warp drive design and wasn't in use prior to his warp flight

This hypothesis is predicated on the fact that the explicit intent behind Zefram Cochrane, whether or not this has been successfully communicated on screen, is that he is the inventor of warp drive as we commonly know it in Star Trek, and that prior to his invention, the existing warp-capable galactic powers were utilizing some other means of generating warp fields.

Citing Ron D. Moore after First Contact was released,

"Certainly Cochrane is credited with the invention of warp drive as we know it in Trek, so we could assume that the Vulcans were using something else – possibly a variant of the contained singularity used by the Romulans. That might have been a much more dangerous and inefficient technology which was quickly abandoned by most of the galaxy when Cochrane's system was introduced."

Now, let's look at some Vulcan ships...

The pre-Cochrane D'Vahl-type starship: This is the type of starship that rescued the survivors of the Vulcan survey ship in Carbon Creek. It is also the type of ship that patrolled above the Forge on Vulcan (as seen in the image). These are warp capable but possess what look like only rudimentary nacelles. The glowing bits look more like impulse engines to me.

The pre-Cochrane Vulcan survey ship: Similar in design to the D'Vahl, with no obvious warp nacelles despite being warp capable.

The pre-Cochrane T'plana-Hath: this ship again possesses no obvious nacelles of any kind, and has what appear to be some kind of engine bells or drive units that angle downward for landing.

The post-Cochrane Vahklas-type starship: this is the only Vulcan starship I can think of that possesses what could be argued to be more typical warp nacelles instead of an annular drive. Obviously the ring shape is hinted at, but it looks like it's only an aesthetic choice here.

When the NX-01 Enterprise encounters a Vahklas-type ship in 2151, T'Pol states that they had not been in use for "a long time."

My hypothesis is that the Vahklas-type represents the first (or one of the earliest) Vulcan attempts at adapting the Cochrane-style warp drive to their own vessels, perhaps utilizing the partial ring shape to increase efficiency over Cochrane's nacelle design, which they consider inefficient.

I think it was this continued lack of preferred efficiency that led the Vulcans to continue working on adapting Cochrane's design for their own purposes, eventually leading to the development of the annular warp drive, possibly sometime between the 2070s and 2100. Definitely prior to the dedication of the Warp Five Complex in 2119.

Presumably, the Vulcans came to the humans with their design if only to show them how their warp drive had ultimately proven inefficient and could be, at least by Vulcan standards, drastically improved upon. This led to the development, perhaps not by United Earth Starfleet or UESPA, of the USS Enterprise XCV-330, the only known human attempt at annular warp drive design.

Ultimately, the annular warp drive proves highly efficient but also highly resistant to course corrections and maneuverability. While this trade off is acceptable to the Vulcans, who do not place an emphasis on exploration, it is antithetical to the very purpose of Starfleet. The design is thus quickly abandoned as a technological dead-end, in favor of the homegrown, For Humans, By Humans nacelle design originated by Dr. Cochrane.

Vulcans, meanwhile, continue to favor the annular warp drive for their own ships, and the High Command quickly adopts it.

This leads to the development of the Suurok-class starship by at least 2136 (when Captain Vanik says he took command of the Ti'Mur in ENT S1E8 Cold Front).

The success of the Suurok-class leads to the further development of the D'kyr-type starship.

Throughout the 2100s, Vulcan starships of all types are designed with the annular warp drive, including ships as small as shuttles and transports.

Over the next century, there would be refinements and adjustments of the annular warp drive design, leading to some slightly different but still ultimately hoop-shaped implementations, as in the 23rd century T'plana-type starship and the early 24th century Apollo-class starship as exemplified by the starship T'Pau in TNG Unification I/II.

But ultimately, the ring-shaped annular drive as developed in the 22nd century remains the favorite, lasting well into the 24th century largely unchanged, although with the addition of what appear to be some slightly more traditional nacelle-like elements, as seen on the Sh'vhal-type starship on Lower Decks.

Thanks for coming to my TED Talk.