Can I just build a stud wall partition between then new utility room and
the
garage?

Yes. However, it must be fireproof. I would use 2 layers of 12.5mm
plasterboard with offset joins. I'd even consider using Fireline board.

I am assuming this would need to be a fire door and also the new partition
needs to be fire proof as well?

Yes. As well as the fire proof wall, you would need a fire door and ensure
that there is an intumescent layer on the door or frame to prevent leakage.

I would need to raise the floor of the garage to meet the existing kitchen
floor.

No. You would need to raise the floor of the utility room to meet that of
the kitchen. The floor of the garage MUST remain lower. It is part of the
building regulations that garage floors are lower than habitable rooms and
kitchens so that any heavier than air fumes and spills stay in the garage.

Christian.