How do I get Bella to stop getting up at 4am?

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We have had her since Saturday, she came directly from her previous home to us through a rescue. She is an adorable little dog and fitting in really well apart from the early wake ups anyway.
At her previous home she slept in her basket in the kitchen. We have her basket but have it in the living room because the kitchen is tiny and I need to shut the door or else she is straight upstairs to dd's room.
She obviously has slept in her basket because she goes in it if I say bedtime and she is fine until 4am and then she starts scratching the door to come out.
It's complicated because ds (and dd to some extent) don't sleep very much and so she can most likely hear them pottering about (the sleep situation is a longstanding issue)

I haven't been going in to her because I don't want to reward her getting up at that time and tbf she isn't distressed she just wants our company I think.
This morning I got up at five and pottered about in the kitchen until six when I went into her (six o clock is the time I am happy for her to get up). She was quiet five til six because she knew I was in the room next to her.
We have a good routine, I get up have a cuppa, make dd's breakfast then about half six take Bella her walk for 45 minutes to an hour. 9 am she has breakfast so she's not getting up for food.
Any ideas please? Oh and the other one is there any way I can stop her going upstairs without fitting a gate. Eric just knows but I think Bella might have gone upstairs at er previous home.

For sleep have an alarm that you set for the time you want her to wake. This will get her used to associating the alarm with morning. I don't use one all the time but because I do for probably about 80% of the mornings I can see my dog waiting for it to go off.

Also check that the light isn't waking her (black out blinds) and maybe consider a white noise machine or fan noise that can help mask external sounds for her.

9am is quite late to be feeding her unless that's what she's used to? Mine is starving when he gets up at 7am and wouldn't take too kindly to being made to wait several hours!

Otherwise as PP has suggested - blackout blinds! But if there's noise going on and people awake she's going to want to be part of that. Dogs are sociable and if mine's awake he has to be with us! If he's asleep he's perfectly content to be left wherever.

Mine very often doesn’t get fed until 9ish and he’s always up by 7 with dh . I think the issue is that this dog is probably use to getting up with the first person in the house who gets up and she can hear your children . Could they not just let her out of the room when they wake up even if that means you need a gate to stop her going upstairs .

She doesn't seem hungry and she was fed after school run at her othe home so maybe even later than nine,I'm pretty sure it is company she is after tbh. She is pretty needy but I suppose she has been uprooted from everything she knows so it's not surprising.
There are blinds and blackout curtains so it is dark in there when she is getting up.
I didn't want a gate because ds and dd can't open them but if there is no choice then they will just have to climb over I suppose. I am having some success keeping her downstairs today (leaving the door open whilst ever she doesn't try to go up) so maybe in time she will learn not to go up like Eric which would mean the door wouldn't have to be closed at all.
I can't complain really she is doing brilliantly after a big upheaval and she is a really lovely dog.