Are there any differences in nuance in these sentences? (c) sounds general-purpose, (a) more like literary expression, and (b) kind of a roundabout way to me.

(a) No sooner had they sat at the table than they started quarreling.
(b) They had hardly/scarcely sat at the table when they started quarreling.
(c) As soon as they sat at the table, they started quarreling.

See a translation

(C) is the most natural everyday way of saying this, although "fighting" or "arguing" are more everyday ways of saying "quarreling," which is not a word people would typically use in everyday speech.

(B) also makes sense but it wouldn't be commonly heard phrased that way, especially "scarcely" which I'm not sure can even be used in this context. If you say "hardly" it would definitely make sense, but be less commonly said than (C).

(A) kind of hurt my head to read. It may be possible to say it that way, but this would be highly literary or poetic language that nobody would say in everyday speech.