As I'm sure your OT and SLP have told you, there's a lot of natural variability in when kids hit language milestones, and late language emergence is not always cause for concern.
Your child --- with no expressive vocabulary at all at 3 years --- is an unusual case, though, which means it's hard to calculate statistics on trajectories for children with his profile. His situation is rare, so we just don't know that much about it, I'm afraid.
That said, there are well-documented cases of children who are classified as "late talkers" but then eventually end up with language ability that is well within the normal range for their age --- these children are sometimes called "late bloomers" and may make up as much as 75% of late talkers (see citations here for studies on this phenomenon). Some late talkers don't end up "catching up" to their peers, though, and may eventually be diagnosed with Specific Language Impairment (if there are no signs of developmental delays in any other areas) or another diagnosis.
Here's a brief overview of several relevant studies, taken from this article (the numbers in square brackets below refer to other studied reference in the article --- to find those studies, open the article and scroll down to the references list at the bottom):
''Late bloomers'', that is, children who make good progress in language
after a slow start, are well-documented in the literature. For
instance, one study followed 26 children aged 2 years old, who were
recruited because their parents reported that they understood complete
sentences but could say only a few words 7. Five months after the
initial assessment, around one-third of the children still had
problems, one-third had made some improvement, and one-third were in
the normal range. Another study followed 10 children who scored in the
bottom 10% for expressive vocabulary at the ages of 18 to 29 months
[8]. One year after initial assessment, six had 'caught up', but the
remaining four still had delayed language. Similar figures were
reported by Rescorla and Schwartz [9], who followed up 25 boys who had
specific expressive language delay when first seen at 24 to 31 months
of age. By follow-up at 3 to 4 years old, ten boys no longer had
impaired language and two of these were above average in their
utterance length. However, the remaining 15 children still had
significant language delays. Although the proportion of late talkers
who have clinically significant language impairment appears to decline
with age [10], some children fail to catch up, and still have
persisting problems well into middle childhood: for instance, Moyle et
al. [11] reported that 37% of late talkers were receiving speech and
language therapy (SALT) at 5 years of age, and Rice et al. [12] found
that around 20% of late talkers had language impairment at 7 years of
age. Measures of morphosyntax appear to be particularly sensitive in
revealing persisting language deficits in late talkers [12,13].
So as you can see, a substantial proportion of children with late language emergence end up not having any language impairment a few years later, but that doesn't necessarily tell you anything specific about your son's case.
There are a couple important things to consider (and talk about with your OT and SLP) when trying to figure out what your son's trajectory might be like:
First, does he show impairment in any areas other than expressive language? The late talkers and late bloomers studied in the scientific literature are cases where children understand language well, they just don't speak (or have trouble speaking). These children score within normal ranges on nonverbal intelligence tests, and they lack symptoms that might indicate another disorder (such as ASD). If your son has a broader developmental delay, then the studies on late talkers, SLI, etc. are not relevant. Instead, you'll want to consider the specifics of language development for children with his diagnosis (which again, I'm sorry to say, can raise more questions than answers --- in children with ASD, for example, there is huge variability in language trajectories and eventual outcomes).
The other thing to consider is the severity of his language impairment and the trajectory of his learning over time. As many as about 15% of 30-36mos have late language emergence (see this link for citations), but many of those children do say some words, just substantially fewer than is typical at that age. So right now, your son is probably at the extreme end of the distribution for late talkers.
In the publicly available data at WordBank, there are vocabulary scores from more than 5000 children 16 to 30 months old, and there are 0 children with no productive vocabulary at 30 months (suggesting that the prevalence is less than 1/5000). You can explore the data on the site, or download it to analyze on your computer. Here is a plot of deciles of the size of productive vocabulary (number of unique words spoken) from 16 to 30 months:
You can see that at 30 months, only 10% of children have less than about 225 words in their productive vocabulary, according to these data (note the disclaimer on this page, though, emphasizing that this may not be a representative sample of the population). I downloaded the data myself to more closely examine the children at the high end of the age range. In these data, only about 1% of children 28-30mos say fewer than 50 words.
Late bloomers typically catch up to their peers around age 3-5, so your SLP will probably be watching your son closely for improvements over the next several months. If he starts speaking, he may still have a small vocabulary for a while relative to his peers, but his rate of vocabulary growth (not just the actual size of his vocabulary) may give you an indication of whether he's likely to ever catch up or if he will retain a language impairment (see relevant citations here).
tl;dr
Your son's situation is quite rare, which is why there are no statistics available to help you predict when he might begin speaking, if at all.
If he doesn't show signs of developmental delay in areas other than speaking, then he might be a "late bloomer" (a child who starts speaking late but ends up without any remaining language impairment). If so, the literature suggests that he'll catch up quickly over the next year or two. There's no way to really know for sure, though, other than to wait and see.
