Nibbling can also be called beaking. Like human babies budgies use their beaks as a means of testing the safety of their environment. This sounds like a little expression of fear when the budgie starts to nip. Try different ways of approaching your budgie with and with out millet. just letting the budgie run across your hand held very still is a good way to reduce the threat.
I cheat a little, by using a show cage. Prepping birds to stand tall for a show judge requires some patience and I have found that it reduces the time needed to step up etc. The show cage has 2 perches. I lay my hand across the perches sit still and wait. the budgies will eventually jump up to the perch from the floor of the cage to the perch. No way budgie gets to perch with out feeling my fingers and hand, after a while the bird forgets my hand is there and is jumping all over the hand and beaking or nibbling the fingers which are like soft perches. Eventually I add a piece of millet and begin praising when they jump to my hand when I say step up. The point is to get the bird to feel safe with the feel , taste, smell of the hand and arm.
I usually train one or two at a time or use an experienced bird with the new guy, If you are using a larger cage like a home cage. You will do the same thing except the bird has more room to avoid contact with your hand. This can take a little longer. Any bird can be desensitized at any age We do similar work with older breeder birds we bring into the aviary. Every body gets to learn their step ups and to come to a dowel in case They get caught in a difficult place and need to be rescued. Be confident in yourself and your bird will understand this. It makes the bird feel more confident and at ease.
Blessings, Jo Ann