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After some advice please (see picture also)

I have an integral garage with a bedroom above. The roof of the house is kind of split level (not sure of the professional name for it) and the bedroom has a dormer window.

The bedroom (my daughters) is so cold overnight and when the heating is off (gets down to 10 degrees often) and the 4th bedroom (used as an office) is similarly cold at the back of the house.

Above the garage door is a vented soffit. I was wondering (or if this would be a bit far fetched) if the cold air would be gusting through the ceiling/floor void and making both rooms cold? The floors of both rooms are noticeable cold under foot.

I am looking into having the garage ceiling over boarded with insulated plasterboard and having the bedroom above redecorated but wanted to sort out the cold first.

Any advice welcome.

Hi adamed13
Most likely problem is that whilst there is insulation directly above the garage ceiling there will be a ventilated gap above the insulation under the bedroom floors.
Even if you overboarded the garage ceiling as you propose, the ventilated gap would remain. Possibly the easiest solution is the insulate above the bedroom floors and top with new flooring. Accepting a slight reduction in headroom.
Regards S