Is there any difference in plain meaning or nuance between "At the start, ..." and "In the beginning, ..."?
The main baggage that "In the beginning" carries around that "At the start" does not involves prominently positioned wording in the Bible, in both Genesis:
In the beginning when God created the heavens and the earth, the earth was a formless void and darkness covered the face of the deep, while a wind from God swept over the face of the waters
and the Gospel of John:
In the beginning was the Word, and the Word was with God, and the Word was God.
That's a lot of baggage. It's certainly possible to read the words "In the beginning" without feeling the resonance of the biblical wording, but the connection is strong in many people's minds.
In isolation, "In the beginning," "At the start," and a third option, "From the outset," have very similar meanings. In a usage note, Merriam-Webster's Eleventh Collegiate Dictionary (2003) offers this comment on the difference between the verb begin and the verb start:
BEGIN, START, and COMMENCE are often interchangeable. BEGIN, opposed to end, is the most general [example omitted]. START, opposed to stop, applies esp. to first actions, steps, or stages [example omitted]. COMMENCE can be more formal or bookish than BEGIN or START [example omitted].
Merriam-Webster's treatment is reasonable, as far as it goes—but what about the argument that start isn't always the counterpart of stop, but is sometimes the counterpart of finish. In that case, the distinction between beginning (vs. end) and start (vs. finish) seems very fine indeed, if it can be grasped at all. I guess you could argue that start is slightly more connected to completion at the other end, and beginning to cessation of any kind—but I seriously doubt that most English speakers go that deeply into their word-choice options before selecting one or the other.
The good news is that whether you say "In the beginning," "At the start," or "From the outset," the vast majority of English speakers will have no trouble understanding what you mean.