Humans are the dominant race of the Nentir Vale anywhere outside of Hammerfast and the Harken Forest.
Halflings are the second most numerous people in the Vale, like humans can be found nearly anywhere.
Dwarves can be found in the towns. For 5th edition, I’d put a small number of hill dwarves in Fallcrest and the mountain dwarves to the east in the city of Hammerfast.
Dragonborn are rare sights in the Vale. A few can be found in the towns of the Nentir Vale acting as bodyguards or caravan escorts. There is also a large group of Dragonborn called Dythan’s Legion who search for past relics from their old Arkhosian empire.
Tieflings are rare and most likely to be found in Fallcrest, living under the thumb of two tiefling families or operating as shopkeepers. Tieflings have a long history in this world and were originally human nobles before they built the now collapsed Bael Turath empire.
Wood Elves are rarely seen in human dominated communities. Most likely a wood elf would be from a small village in the Harken Forest.
High Elves are rarely seen in the Vale at all. In 4th edition, Eladrin were rare in the Vale, so we had our Eladrin come from the Feywild. For 5th edition, I would have a High Elf come from outside the area or the Feywild, but a better option might be to swap High Elves for the Eladrin build used as an example in the 5th Edition DMG.
Half-Elves can be found in small numbers, but due to the scarcity of elves in the Vale the interaction with humans required to create a half-elf would be rare. You can decide on your own how society treats the offspring stuck between two worlds.
Gnomes are almost non existent in the Vale. I’d place tinker gnomes in the towns and forest gnomes living in communities under the trees of the Harken Forest. If you want to create a community of Tinker Gnomes, go for it, but a more likely story is that the gnome traveled to the region.