How to use USD OIS discounting for local currency uncollateralised swaps?

I am wanting to do MTM valuations of uncollateralised swaps as our banks have switched to using the USD OIS curve for their discounting (assuming no xVA adjustments). None of the cashflows we are using are denominated in USD (our local currencies don't have liquid OIS curves).

How do you make the adjustment for currency basis between the USD and native cashflows? Does anyone know how this extends to cross-currency swaps that are both in different currencies to the curve?

We can't value the swaps at its own funding rate because we have no idea what that is - the bank is providing MTM calculations with no adjustments so I presume we can do the same (base discounting using the OIS curve).

Any light on any of this would be seriously appreciated.

Suppose you wanted to value a 5Y EUR IRS with a USD cash collateralised curve this is the broad process:

Get the 5Y EUR 3M / OIS basis, say this is 10bps: This establishes the discounting basis in the local (EUR) currency.

Now get the 5Y EUR/USD Cross-currency basis, say this is EUR 3M-IBOR - 40bps: This establishes your link to dollars.

Now get the 5Y USD 3M / OIS basis, say this is 20bps: This establishes your discounting principle in USD.

How many basis points has it cost you to transform EUR OIS into USD OIS? +10 -40 -20 = -50bps. Another way of looking at it is if USD 3M flat is EUR 3M -40 according to the cross-currency basis, then making the visual adjustments for the OIS curves in their respective currencies means USD OIS Flat is EUR OIS -50bps. This means USD is the most expensive to deliver, which it is in current markets.

Now if the NPV of the EUR swap is EUR 100,000 as measured in the local EUR OIS, then the difference when the new discounting methodology is applied will be multiplying the discount risk by 50bps.The effect will be positive since the new USD-OIS curve is lower than the EUR-OIS. Roughly discount risk on a swap is: $$DiscountingBasisRisk = \frac{PV}{10000}*\frac{tenor}{2}=25 EUR$$ So in this case the change of discount curve bumps the NPV up to 101,250 EUR, with a pinch of salt for different curve shapes and other minor effects.

For more detail see Darbyshire: Pricing and Trading Interest Rate Derivatives.