The royal family would not have acknowledged their child at all; no HRH and probably no title. Maybe the Lord or Lady that comes with being the child of a Duke, but even that might be up in the air. Their kid probably would have considered themselves an American and not really a royal or member of the House of Windsor. Or I’d like to hope Wallis would not have raised some tacky fake royal Real Housewives kid who would go around making everyone call them HRH and claiming to be the rightful monarch. You never know though. I think David would have been an amazing dad and spoiled his kids like Bertie did. W&E’s imaginary kid would have probably gone into the industry in some way, like acting, modeling, a band, or just being an “it girl/guy” in NYC or Paris. I just could not for the life of me imagining those two having a child who just went to college and got a white collar job and moved to Connecticut or whatever. If the kid/kids took after David and were attractive they would have all kinds of cool options open to them.
They would not have any relationship with the royals except maybe the occasional wedding/funeral. They probably would not have been invited to W&K’s wedding which would have caused a ton of tabloid speculation.