DD was born on 26th December. We always have her birthday on 26th, with cake (and usually candles - it's a sensory thing that some years she's already on overload so candles can cause a meltdown if done wrong, so we may not).
DD gets to choose her cake in M&S on Christmas Eve most years, although a couple of years I have actually baked it at home (while I do bake, Christmas is normally chaotic enough to have me wanting to not "have to" bake on Boxing Day). We usually have some sort of gathering, either at home or in the house we have rented (if we travel to our DPs and DPILs at Christmas, we rent a cottage normally) - or in on of our DPs/DPILs houses if necessary (we try to avoid that one though, as there are already 2 matriarchs there and neither wants to accept the other's hospitality....).
If we are "down home" (i.e. visiting extended family), then it is a slightly more formal party time - specific start time although can be very staggered departures between DGPs staying a couple of hours and DAunts/DUncles settling in for the evening.
If we are in our own house, we have an "open house" afternoon for neighbours and any family who will travel. Mulled wine and nibbles, and DD gets cake at a suitable point. Some neighbours tend to come early afternoon and stay for an hour/2, while others have family commitments and come in later (time varies depending on how their extended family meal goes) but can stay very late (very relaxed!). DD gets her presents from us in the morning then, and cake some time in late afternoon/early evening.
And we always give a birthday present separate to Christmas presents, and a card. It can vary with others - some joint, some separate, some only 1 (for 1 or other), and some do different things different years.