Personally I’d argue they are British, not simply English… They have British passports, not English ones. They are British citizens not English citizens. I know that some people feel more English, Welsh, Scottish or Northern Irish than they do “British” (and that’s totally valid) but I’d be surprised if the “British” Royal Family feel they’re English over British considering how much time they spend in Scotland… Being English, Scottish etc over British is a personal feeling. Most people born in England feel British over being English. In the other countries that’s reversed. But bluntly, at the end of the day, they are British Citizens, and the UK is one nation. Maybe one day there will be independence and then the conversation will be different.
Also, they might have ancestors who weren’t born in the UK, but they were all born in the UK (apart from Lilibet) so they’re British/English whatever you want to say.