HDR is all about the color accuracy and contrast. While LCD has, for years, been very good at colors, the contrast ratios have always been greyer than what plasma has delivered. This is where OLED has a real advantage in meeting HDR requirements. They are able to reach down and really hit the black levels which is required of HDR displays. What they may lack, is enough brightness to hit the peak whites which HDR is asking of them, but I don't think this will be a long term issue. I think that OLED is bright, and can get brighter, while still maintaining those inky blacks, and delivering colors that are going to rival what LCD displays can deliver.
There are other OLED issues that I'm not about to discuss, but when it comes to HDR, OLED has a solid advantage.
Oh - and HDR photography and HDR displays aren't the same thing. It's not 'processing' - it's an actual measurement of the capabilities of the display which includes color gamut reproduction, peak brightness, and black level measurements.