I go with a modified version of the Decipher/LUG explanation:
The Romulans had warp drive quite early on, but did not possess antimatter technology. Instead, their system relied on the single-shot magnetic bottle drive (described in Way of D'era) and the singularity drive for very large ships. In addition, smaller ships would often be carried by larger, singularity-equiped ships (I got this idea from one of the early novels) - amongst other things, it neatly explains the design of the D'deridex (for me at least!).
They got antimatter drives from the Klingons in the famous tech exchange.
OTOH, another (and very likely) interpretation is that the bird of prey in "Balance of Terror" had its warp drive powered down because otherwise it would compromise the cloaking device. In this scenario, they didn't dare power up the drive as the Enterprise would have caught them more easily. They also may have been aware that they would not have been able to outrun a Connie at warp. The singularity drive is simply a later development.
ymmv
Jon
"There are worlds out there where the sky is burning, where the sea is asleep and the rivers dream; people made of smoke and cities made of song.
Somewhere there's danger, somewhere there's injustice, and somewhere else the tea is getting cold. Come on, Ace, we've got work to do."
THE DOCTOR, "Survival" (Doctor Who)