I cannot give a thorough answer right now (if I even find one) because I'm on my phone, but here are some papers that could inform the question in the meantime (or their abstracts at least!)
One element I have gathered so far is that the evolution of nervous systems likely involved not the evolution of wholly new molecules and structures, but the cooptation of existing structures for new purposes. Meaning the first neurotransmitters would have been pre-existing with some other purpose in early proto-neurons, not new molecules evolved expressly for the first synapses. One paper talks about neurotransmitters like dopamine being common across all life, including plants (no nervous systems there!). This does not answer which of these molecules would have been the first neurotransmitters however.
The abstract of this paper doesn't say much but the title is tantalizing:
"New Trends and Perspectives in the Evolution of Neurotransmitters in Microbial, Plant, and Animal Cells"
This paper shows a unicellular eukaryote reacting to various neurotransmitters. If the full paper goes into the phylogeny some more it might be interesting:
"The Phylogenetic Background of Neurotransmitters in the Unicellular Organism Tetrahymena Pyriformis"
This paper's abstract says all I was saying about the development of neurons and synapses relying on pre-existing molecules and structures:
"Evolution of Neurotransmitter Receptor Systems"
What I'm looking for to find how close science is to answering your question (or if it already has) is papers looking at the phylogenetic relationships of different neurotransmitters and related molecules. That might say a lot about which ones have been used as neurotransmitters the longest. I haven't found this yet but I'll try more tomorrow when I'm not on mobile.
ETA: will re-edit this comment later, but this paper answers your question I believe, or as well as any could at present at least:
"Evolution of Animal Neural Systems"
This paper, which is available in full and is a review paper from 2017, looks at the evolution of every aspect of animal neural systems (i.e. nervous systems mediated by neurons, a concept the paper also defines because the line is apparently blurry). One interesting aspect of it is that while you point to Cnidarians as the most "primitive" nervous systems, the paper points out the latest evidence suggests Ctenophores are the earliest branch off the animal tree, meaning nervous systems either evolved convergently, or sponges lost their nervous systems secondarily, in which case Cnidarians would lose this special status.
The paper has a section about neurotransmitters, which says the following:
Several types of molecules are used as neurotransmitters; their evolutionary deployment in different synapse types across animals is fascinating and still poorly understood. Many are used widely in eukaryotes for intercellular communication, but some of the biogenic amines may be present in animals as a result of the late horizontal transfer of synthesis enzymes from bacteria(Iyer et al. 2004). For instance, epinephrine and norepinephrine are important neurotransmitters in vertebrates but not in protostomes (but see Bauknecht & Jekely 2017), whereas the opposite is true of octopamine and tyramine (Figure 4). Cnidarians make a set of neurotransmitters similar to those in vertebrates (Kass-Simon & Pierobon 2007), but Nematostella expresses most nonpeptide types in the endoderm near the pharynx and testes—only peptide transmitters are found in neurons(Oren et al. 2014)
Intriguingly, ctenophores seem to use a much more restricted set, as glutamate is the only well-validated neurotransmitter (Moroz et al. 2014). This is consistent with the theory that neurons arose independently in ctenophores and planulozoans because vertebrates and most protostomes use acetylcholine at the NMJ [neuromuscular junction I assume -my edit]. However, arthropods use glutamate at the NMJ, just as ctenophores do ( Jan & Jan 1976), and cnidarians probably use neuropeptides (Oren et al. 2014). Although sponges do not have true synapses, they use γ-aminobutyric acid (GABA), glutamate, and nitric oxide to coordinate contractions (Elliott & Leys 2010). Trichoplax individuals also lack synapses,but their secretory flask cells label for FMRFamide, suggesting a conserved role in transmission for this peptide class (Smith et al. 2014).
This gives at least a few names of neurotransmitters that are particularly conserved and candidates for neurotransmitters the common ancestor of neurons used (glutamate, acetylcholine; they may have been peptides, whatever that covers), and I think makes it clear that it is very much not known what those first neurotransmitters were.