Has the nanny been with you prior to the baby?
New baby plus someone else taking your place (they had your undivided attention for how long - such as did you take a lot of maternity leave?) will result in a child feeling rather left out.
Have you pointed out to the nanny that they are employed to care for BOTH children, not just to hold the baby?
How much contact time is there on a typical day between the nanny and your DD? As age 4, she may be at school.
What are the typical things that nanny does with the children, for example does your nanny take them out to places, swimming, toddler groups, visiting friends of your DD?
I nanny for a baby (4mths), 3yr old and 6yr old. Yesterday I spent time with 3yr old whilst baby took a nap... we charged around the garden, 3yr did some bouncing on the trampoline, did some digging and watering in the garden (I suggested we should plant some strawberries soon - think it's about the time of year to do that), and did some drawing/colouring.
Babies can be quite demanding at times but when they are not being demanding, then that time should be spent as much as possible with the older child. Is your nanny not doing that?