In OED sense B1 1a of the noun/adj thick it says the following with examples. No indication is given that it is uniquely British, but then none of the examples are American. I suspect it may have something to do with its origins being a metaphor from fog.
b. fig. The position, time, stage, or state in which activity is most
intense; the midst, the height (of an action). Always in the thick of.
1681 J. Flavell Method of Grace ix. 214 Something they enjoy..in
the very thick of troubles.
1821 Byron Sardanapalus iii. i. 86 Where a soldier should be, In
the thick of the fight.
1849 C. Brontë Shirley I. i. 13 They are in the thick of a
1870 J. H. Burton Hist. Scotl. to 1688 V. lv. 348 The bishop was
in the thick of these splendid projects.
1885 H. Dunckley in Manch. Examiner 15 June 6/2 We are now in the
thick of a Cabinet crisis.