A new baby takes a lot of time for mum and dad and also friends and relatives come and think the world of the new addition, so we all understand why the older sibling(s) feel put out. It is perfectly normal. Only very few children do not try tantruming as a method of control, and just because your child has or has never tantrumed does not make anyone a worse or better parent.
Give children as many choices as possible from things the parent or caregiver has pre-chosen. That way the child get some control in their everyday life. It can prevent tantrums over time.
This is the why -- negative behaviour is always noted and so the tantruming or naughty child is given attention, and any attention is better than no attention. This doesn't have to be logical. S/he may in fact be getting plenty of attention, but there's jealousy tipping the scales in the eyes of the child. She knows exactly how to make you 'see' her.
Nanny is probably experienced and never makes an idle threat or changes her mind or gives in once she's made a decision. It is easier for her. She has time away from the situation, breaks, hours off, she doesn't worry about getting the child to an appointment the way a parent does/has to. Mums and dads do not get time off unless they can do it for each other.
The child may love the nanny, but this is not the same love as parental love. This is why children will listen to nannies/caregivers or grandparents or aunties over mum some of the time. The child knows she owns her parents. (That is actually a very good thing.) The 'upside' of these tantrums is that she trusts you and feels safe enough to have one. (Yay!)
Tantrum Prevention Tactics from WebMD
Instead of having to stop a temper tantrum after it starts, prevent it
by following these tips:
Avoid situations in which tantrums are likely to erupt. Try to keep
your daily routines as consistent as possible and give your child a
five-minute warning before changing activities.
Communicate with your
toddler. Don't underestimate his ability to understand what you are
saying. Tell him the plan for the day and stick to your routine to
Allow your child to take a toy or food item with
her while you run errands. It may help her stay occupied.
your child is well rested and fed before you go out so he doesn't blow
up at the slightest provocation.
Put away off-limit temptations (for
example, don't leave candy bars lying on the kitchen counter close to
dinnertime) so they don't lead to battles.
Give your toddler a little
bit of control. Let your child choose which book to bring in the car
or whether she wants grilled cheese or peanut butter and jelly for
lunch. These little choices won't make much of a difference to you,
but they'll make your child feel as though she has at least some
control over her own life.
Pick your battles*. Sometimes you can give
in a little, especially when it comes to small things. Would you
rather let your child watch 15 extra minutes of television or listen
to her scream for 30 minutes? Distract**. A young child's attention is
fleeting and easy to divert. When your child's face starts to crinkle
and redden in that telltale way, open a book or offer to go on a walk
to the park before it can escalate into a full-blown tantrum.
Sometimes, humor is the best way to distract. Make a funny face, tell
a joke, or start a pillow fight to get your child's mind off what's
upsetting him. Teach your child other ways of dealing with
Children who are old enough to talk can be reminded to
use their words instead of screaming.
Praise your child for getting it
right. When he stays cool in a situation that would normally have
triggered a tantrum, tell him he did a good job of controlling his
*Give in immediately if you are going to ultimately give in. The longer you make her fight, the longer she is willing to fight the next time if you've shown her you will give in. So if you must be somewhere or do something and have no time for a tantrum -- give in right away. Don't even say. "No."
**Redirection is magic. It is a method of parental behaviour management that simply changes the subject. You redirect the child to something else. It is something you work on together until you get this method working.
(I added the bold type and the notes (*/**).)