The comparison between the three years of training in Daniel 1.5 and Nebuchadnezzar's second year in Daniel 2.1 creates a difficulty only if the indications are interpreted according to a modern reckoning made up of full years counted from the day of accession.
The Babylonian system, by contrast, distinguishes the accession year from the sovereign's first official year.
A training begun during Nebuchadnezzar's accession year in 605 BC can thus cross his first year and end during his second official year.
The three years of Daniel 1 can correspond to portions of three calendar years without necessarily representing thirty-six months.
This reconstruction makes it possible to preserve the natural progression of the account: Daniel is trained, examined, admitted to royal service, and then confronted with the crisis of Nebuchadnezzar's dream.
It also explains why Daniel can be affected by the decree against the wise men in Daniel 2.
The solution must nevertheless not be presented with a precision that the texts do not possess. The exact months of Daniel's arrival, of the beginning of his training, of his examination and of the dream are not given.
The most prudent conclusion is therefore that Daniel 1.5 and Daniel 2.1 can be chronologically compatible if one takes into account Nebuchadnezzar's accession year and the possibly inclusive nature of the counting of the three years.