Having had some first hand-experience, I can give you more detailed information:
In both of these regions, there are still some signs that have the Georgian name written on it, but almost all of them have been put up before 1989.
The language situation in those two regions is different.
Abkhazia is a multi-cultural country and Russian is the language spoken everywhere, although Abkhaz is also learnt and spoken by non-ethnic Abkhaz people (although fluency is a very different matter…) Most Russians and Armenians living in Abkhazia have a positive attitude towards Abkhaz language and Abkhaz have a positive attitude towards Russian.
Georgian is only spoken by members of the Georgian minority and some older Abkhaz people who were forced to learn the language during Soviet times and often refuse to speak it now.
The only part of Abkhazia where Georgian is still the most widely spoken language is Gali region, where most Georgians live. But even in this part of Abkhazia, most signs on shops are in Russian while official signs are mostly bilingual in Abkhaz-Russian. Still, you have a good chance of finding at least some Georgian signs.
Tkvarcheli has a significant non-Georgian population and hardly any non-Georgian is willing to or able to speak Georgian, so many Georgians will speak Russian in public. Most schools there teach in Russian as well and younger Georgians tend to be more fluent in Russian than in their mother tongue.
Outside of those two regions, Georgian is not really used anymore at all and less understood than, let’s say, English.
Modern street signs in Abkhazia look like these:
(Abkhaz name, Russian name below)
South Ossetia today is ~90% Ossetian, so the Ossetian language has a stronger position than Abkhaz has in Abkhazia. Russian is spoken by nearly everyone as well, but on the streets, you will mostly hear Ossetian. Most street signs are bilingual in Ossetian and Russian. Few ethnic Georgians remain in South Ossetia, although there are a few villages that remain mostly Georgian. In those places Georgian is still spoken, but as in Abkhazia, even in those villages most signs are in Russian as Ossetians coming by generally don’t understand Georgian language and have a very negative attitude towards it.
This is the town sign of Tskhinval, the capital:
Street signs look like these:
(Ossetian name big, Russian name smaller)
As far as I know, South Ossetia was initially mapped in Russian only, but some (or only one?) Ossetian mapper (soslan) then changed the default name to Ossetian only and the Russian name was put in name:ru. As far as I know, Andrey Kozaev is also from South Ossetia.
I don’t know of any local mappers from Abkhazia. Abkhazia was always mapped in Russian, with ulil, who seems to be from Germany, being the most active mapper there.
I think the decision not to use the Georgian name is a purely pragmatic one. I don’t like the way Abkhazia and South Ossetia are treating their Georgian heritage/minority, but nonetheless that’s the way it is. To me, it’d make no sense to ignore reality.
It’d be great to add Georgian translations of POIs and street names in both of these two regions as they’re mostly lacking.