I have actually looked into this a lot and I don't have a full answer.
- You can (sort of) see what the market participants think by looking at the consensus "quanto factors" published monthly by IHS Markit Totem. They even have some term structure, although usually it's the same factor for all tenors. For some sovereigns, people sometimes trade a CDS denominated in local currency referencing hard-currency debt. (This stuff isn't novel, by the way - LTCM used to trade quanto CDS on the likes of Italy back in the 1990s.) The idea is that if the sovereign defaults, then the protection buyer gets paid the notional using the post-default exchange rate. So the CDS spread (market standard quote) is discounted in comparison with the USD-denominated CDS spread by the quanto factor, which is approximately 1-the devaluation on default. (Totem also has quanto factors for many names denominated in EUR and JPY.)
(As an aside, there are other kinds of contracts you could use to express a view on the quanto factor, which you could then hedge with the more vanilla quanto CDS. For example (self-referencing extinguisher) you loan to a credit-risky foreign sovereign their own local currency, with a clause that the loan is torn up (extinguishes) if the sovereign defaults on hard-currency bonds. Or (extinguishing cross-currency swap) you could pay USD to a corporation in another country and receive local currency, again with a clause that if their soveregn defaults on hard-currency bonds, then you extinguish either the local-currency leg or both legs.)
But how meaningful are the consensus quanto factors? It is available for only a few countries. There are usually much fewer than 10 contributors, and they don't actually trade often, just mark their books. I would not be surprised if the quanto factor turns out to be far from the consensus number when a trade actually happens, and even further from reality when a sovereign default actually happens.
You can also try getting indicative quanto factors from inter-dealer brokers like TP ICAP and BGC Fenics MD, with similar caveats.
You could look at the history of sovereign defaults, such as Moody's, S&P, or Bank of Canada CRAG, and try to figure the FX depreciations (noting that some depreciation happens before the default), and take some kind of average. I am skeptical that this would be meaningful, given that every default is very different, and there are so few of them.
You could collect fundamental data like GDP growth, local equities, inflation, and FX forwards, and build some kind of econometric structural model-like calculation. I've seen a few attempts at doing this and was not impressed. (Such models can be used to saisfy a product control group that the quanto factor marked by traders is not entirely made up.)
As an extercise in machine learning, you could collect as much country econometric data as you could find, and country news, correlate it to the historical databases of sovereign defaults and to the quanto factors from Totem, and then you may be able to predict
what the Totem consensus quanto factor would be for a country that's not in Totem
how Totem consensus quanto factor will react to the newly available economic data and news.
I found all these papers somewhat useful:
Ehlers, Schoenbucher The Influence of FX Risk on Credit Spreads http://www.actuaries.org/AFIR/Colloquia/Zurich/Ehlers_Schoenburcher.pdf
Brigo, Pede, Petrelli Multi Currency Credit Default Swaps: Quanto effects and FX devaluation jumps https://arxiv.org/pdf/1512.07256.pdf
M.B.Chernov et al Sovereign credit risk and exchange rates: Evidence from CDS quanto spreads https://sites.google.com/site/mbchernov/ACS_quanto_latest.pdf
Lando, Nielsen Quanto CDS Spreads https://research.cbs.dk/en/publications/quanto-cds-spreads
Manzo, Saret (Two Sigma, focusing mostly on EUR) What Sovereign CDS Spreads
Potentially Tell Us about Currency Risk https://www.twosigma.com/wp-content/uploads/SV_05_17.pdf