I find that between 2 and 3 years old, some children will associate the arrival of a carer, with the departure of a parent - and thus will create a fuss. However, it is very clear often (as others have commented) when the child is trying to manipulate the situation. To me it seems as though it's like the protest cry that children do when they are tired and don't want to go to bed. As soon as the parent is out of sight, the child stops. This is not something that just occurs with nannies - it occurs at pre-school, nursery and at childminders. Some children grow out of it in a short period of time, while others will take a lot longer.
Thus it can be hard to tell if the child and carer get along, if you are only seeing the child with the carer at the hand-over time. Therefore, dropping in unannounced is a good idea.
Golden - given that your twins (I am presuming you have twins, as you say '2 year olds' so more than one) adjusted to nursery within a few weeks, I would expect a similar pattern to have been followed with their nanny. Yet, you are saying it has gone on for 7 months... so it is a concern.
What are they like at the other end of the day? Do they seem happy when you arrive home?
Note: I do not give children a cuddle when I am about to leave... I tend to make a reasonably quiet exit - sometimes one of the children will follow me to the front door (an on occasion a child has protested strongly at my leaving, holding firmly onto my leg and screaming), while at other times I will be ignored. To avoid difficulties with departures, I do it in a low key way... the children know that dad/mum is home, and it's their opportunity to spend some quality family time together.
Sacking a nanny due to the children not liking the nanny - hmm, tricky one.
Probation period in contract with a short notice period, then following probation having a month's notice period - then you can dismiss a nanny for any reason you like, by just terminating the contract giving the appropriate notice. If you need to get an employee out faster, pay them in lieu of notice. Not sure there is much else you could do... it would not be instant dismissal, as I doubt it would fall under the Gross Misconduct section of contract.