I agree with Finman, my experience this season, together with observation and mathematics, suggests that with a small colony the number of nurse bees is the limiting factor in the initial stages.
The queen will only lay as many eggs as the nurse bees can feed and cap. A breeding cycle is approximately 3 weeks.
Each breeding cycle might result in a net increase of 10% - 15% in the overall number of bees.
Arithmetic suggests therefore that the colony size will double approximately every seven breeding cycles, assuming adequate food, good weather, and a healthy queen and brood. Not all of which prevail all of the time.
Therefore it is likely to take 20 weeks or more to achieve a full hive from a small nuc, and longer if it has large frames like 14x12 or bigger (double brood ?)
Therefore starting in early March, full colony size will not be reached in less than 5 months, i.e. August, and only then if there has been no bad wet weather in summer.
With big frames to draw a lot of wax onto, having reached full colony strength the bees will concentrate on putting pollen and honey stores in the brood box during the first season, leaving little or no surplus to be put into supers.
They should do well in their second season if disease is kept low.
This has been my experience this year and I have learned there is no substitute for a large colony at the start of the season.
The best and quickest way to get a 3 frame nuc to develop and grow faster is to add bees and brood from other colonies, or combine more than one colony into one hive.
There is nothing much wrong with starting with a three frame nuc, but it's a slow buildup. Sometimes that doesn't really matter.
All the above assumes no swarms occur and the queen lays all the time, both are difficult things for a new beekeeper to achieve.
