1888, "a scrimmage in rugby," abbreviation of scrummage, a variant form of scrimmage (n.). The transferred sense of "continued noisy throng" is by 1950.
1833, American English, in countrified humor writing of "Major Jack Downing" of Maine (Seba Smith), "stylish, splendid, fine;" probably a colloquial alteration (intensification) of sumptuous. By late 19c. especially of food, "delicious, delightful," and it was noted 1890s and early 20c. as a vogue word among college girls (also as scrum, scrummy). Related: Scrumptiously; scrumptiousness.
OED (2nd edition, print) has scrumptious as probably identical with dialectal scrumptious "mean, stingy, close-fisted," and ultimately related to shrimp. The editors insist the sense transition "is not impossible," and they compare nice.