I've converted my ongoing Blackmoor campaign to D&D5 and here's what I got so far:

Races
- Thonian, High Thonian and Peshwah are simply humans of different ethnicities and there are no mechanical differences between them.
- Docrae are simply a different ethnicity of halflings and should be considered as such.
- Cumasti elves are high elves, while Westryn elves are wood elves.
- Dwarves from the Stormkiller mountains are considered Mountain dwarves while those from the Crystal Peaks are considered hill dwarves.
- There are no dragonborns, tieflings and dark elves from Blackmoor.

Classes
- Arcane Warriors are eldritch knights.

Monsters
- Froglings are bullywugs.
- Killer frogs can be modeled as a swarm of quippers.

Misc.
- A spell focus can act as an arcane implement.

Nice!

I like lazy conversions actually.

Could things like Thonian, High Thonian and Peshwa be handled through backgrounds?

I am still a n00b when it comes to 5E.

Although I generally agree with your decision to leave Dragonborn and Tieflings out, the Blackmoor 4E guide has some ideas in how to integrate these into the Blackmoor setting.

I find your decision to make Westryn plain woodelves quite interesting.

-Havard

I personally wouldn't have an ethnicity as a background, as it can lead to a rash of cookie-cutter characters, but backgrounds can be used to highlight some aspects of the character's culture.

For instance, most Peshwah characters will probably have the Outlander background but nothing should prevent a player from taken a different background. For instance, if your Peshwah character became a mercenary, you could select the soldier background. If he had been captured by the Afridhi, you could create a slave background.

Many High Thonians will select the noble background, but an High Thonian who was abandoned at birth could select the urchin background.

Thonians could be anything.