As you've said they are capable of making both types of taps (High and Low, they've been called) simultaneously, and the lack of a tap can also communicate information, this means there are four possible taps:
High
Low
High + Low
No Tap
You've specified they are capable of making and distinguishing 12 taps per second, which means every second there are 16,777,216 ($4^{12}$) potential combinations (including repeating the same tap 12 times) per second. This is essentially a quaternary system. In comparison, binary (tap or no tap) communication at 12 taps per second would produce only 4,096 potential taps per second.
The smallest unit of information is a single tap lasting 1/12th of a second, but the largest would last multiple seconds maxing out at the limit of the human attention span. Since we're prizing speed above all other qualities, let's assume the longest tap-word is three seconds long. 209
That's over 50 million potentially valid word formations (4$^{12}$ x 3 = 50,331,648).
With over 50 million twelfths-of-a-second slots to be filled, it's entirely possible for this language to compress more meaning into a three-second word (or shorter) than most natural languages manage in complex sentences. Consider that English has roughly a tenth the number of words that your tap-lang could contain.
For an example of a constructed language pursuing higher info density, look at Ithkuil. It is a constructed language designed to convey complex meaning without ambiguity.
The many examples from the original grammar book show that a message, like a meaningful phrase or a sentence, can usually be expressed in Ithkuil with fewer sounds, or lexically distinct speech-elements, than in natural human languages. For example, the two-word Ithkuil sentence "Tram-mļöi hhâsmařpţuktôx" can be translated into English as "On the contrary, I think it may turn out that this rugged mountain range trails off at some point".
So it's entirely possible that in a span of a few seconds your tappers could have conversations that would take us a minute or more. The difference of a few taps (each a twelfth of a second long) could completely change the meaning of a tap-word or phrase, meaning this would be a synthetic language if not polysynthetic:
Polysynthetic languages typically have long "sentence-words" such as the Yupik word tuntussuqatarniksaitengqiggtuq which means "He had not yet said again that he was going to hunt reindeer." The word consists of the morphemes tuntu-ssur-qatar-ni-ksaite-ngqiggte-uq with the meanings, reindeer-hunt-future-say-negation-again-third.person.singular.indicative; and except for the morpheme tuntu "reindeer", none of the other morphemes can appear in isolation.
In conclusion your tappers would "speak" faster than the general population, and in short bursts, in my opinion. 312