How different biologically are Remans, Romulans, and Vulcans? According to the question How are Vulcans and Romulans different from each other?, when Beverly Crusher tried to heal a Romulan with a healing technique that works on Vulcans, it failed on the Romulan, yet only 2000 years had passed since the Awakening (the time of Surak). So, my main question is: How different biologically are Remans, Romulans, and Vulcans?
The short answer is, as different or similar as the writers need them to be for a plot.
But let's try and clear up a few things, specifically about ...
There isn't much in cannon that state the history and genetic heritage of these guys. I've seen a few people say in response to this question that they were the original inhabitants of what is now Romulus, I say show me the evidence, I've yet to find a cannon reference to support this.
The only reference I can find for Reman heritage is the non-cannon book series Vulcan's Soul. In this take on the ST universe, Remans were Vulcanoids who got trapped on Remus at roughly the same time as the first Vulcanoids settled on Romulus. They adapted to the harsh conditions with a little DNA modification using some local bacteria.
Back to cannon and we can only draw the most basic comparisons with Remans and Vulcans/Romulans, and that is they are all humanoid with pointy ears. Vulcans and Remans have also both displayed some telepathic traits.
Vulcans and Romulans
As stated by the OP, Romulans and Vulcans have only been truely seperated for about 2000 years before TNG and still remain very similar in appearance.
Romulans have not displayed any telepathic abilities, but that could just be that they haven't learned how to unlock that potential as it would involve following some of the Vulcan teachings (meditation for example).
So how are there any real genetic differences between the two species? Well at this point we are back to speculation.
We know that Romulans practice some form of eugenics. It was stated in TNG episode "The Enemy" that Romulans would let disabled babies die. So after two millenia they may have eliminated some genes in their make-up that would still be part of Vulcan DNA.
We also know that both Vulcans and Romulans are partial to the occaisional interspecies mingle from the existence of Spock and Yela (the daughter of a Romulan and a time duplicate of Tasha Yar). Given that Vulcans and Romulans held very different territorries for millenia, it is conceivable that there was limited interspecies mingling with different species that overtime left very little impact on the physiology of both species except at a genetic level.
They're nearly identical
In TOS Enterprise Incident, Chekov is tasked with locating Spock aboard a Klingon D7 staffed by Romulans. It takes Chekov several minutes to do this, commenting about how similar they are. He had this trouble even considering that Spock is half human, and thus should be even easier to find than a full-blooded Vulcan.
In the same episode, it seems as if a Romulan Commander was perfectly willing to pursue certain intimacies with Spock (although this means little in Star Trek).
In short, they're just different enough that you can tell the difference with a tricorder when the plot needs it.
If we accept the Awakening period as the time of the split, this isn't long enough for anything resembling speciation to occur, especially between peoples who apparently had at least primitive forms of star travel.
A small amount of genetic drift may have occurred (like that between different subspecies of animals), but it wouldn't be enough to call them separate species or prevent interbreeding (as with Saavik, who was at least originally to have been a Vulcan-Romulan hybrid). Also, as noted, at least with a navigator operating the sensors, even the Enterprise's sensor suite (as of TOS dates) had trouble telling a Vulcan-human hybrid from a (presumably) full-blood Romulan (though Spock did strongly favor the Vulcan side, at least physically).
If the inhabitants of Pitcairn Island (isolated from the late 18th century until the 1930s) hadn't drifted enough to even change their language noticeably (from the mix of Polynesian and English they started with), I wouldn't expect even 15 times as long a separation to make a difference observable without DNA analysis. Further, it is canonical (TOS, "Balance of Terror") that the Romulan language is still similar enough to Vulcan for Spock to translate it with only minutes of exposure. Language generally changes far faster than genes.
As for Remans, I know nothing about them -- but comments suggest they were the original natives of the Romulus/Remus system, a completely different species. In the Star Trek canon, they likely still can interbreed with both human and Vulcan/Romulan stock.
While we can assume that the Romulan split happened around the time of Surak there are other factors to consider. Consider if Earth's population split and one group went off to another planet. If Aboriginal Australians decided to leave in one ship and start a space empire that would falsely imply a longer period of genetic drift if some aliens compared the colonists to what remained on Earth.
Also while one can safely assume non-canon books have and will continue to say the split happened because of the society changes brought on by Surak - we only know for certain Romulans did not accept Surak either because they left before him or they rejected him. Note Earth was sending out colony ships in the 1990s during the Eugenic Wars. It is completely reasonable and implied by Spock that Vulcan pre-Surak was equivalent to Earth during the Eugenic Wars and WWIII. So there could be several Vulcan colonies Pre-Surak and one may have simply lost touch with Vulcan. For example the various planets in TNG Gambit and the Stone of Gol artifact.
We should also assume that visually Vulcans and Romulans are equivalent despite the occasional extreme TNG forehead makeup.
The only indication of a large genetic difference is from TNG "the Enemy" such that a Romulan patient needs an emergency "ribosome infusion" and that the Vulcans on the ship are not compatible. Somehow Worf is.
How an alien could possibly be more compatible is not discussed. So I think the audience should not infer too much from this other than there was more difference than Doctor Crusher would have expected.
Remans did not exist before Nemesis (the second to last nail in Trek's coffin) came out and there is no biological information about them in television. They showed up in an Enterprise episode too (the last nail). Thus there was not a lot of opportunities to learn about them.