From a linguistic and phonetics standpoint, in the majority of cases, the suffix -t as a formative of past participle of verbs appeared as a result of the reduction of -ed to -'d, and the devocalization of d.
Here is the full explanation from OED with some historical notes:
Formative of the past participle in some weak verbs, for earlier -d and -ed (see -ed suffix1), due usually to the devocalization of d after a breath consonant, as in nipped, nip'd, nipt. In some verbs the use of t for -ed goes back to Germanic, esp. in apparently contracted or irregular verbs, as bought, brought, might, thought, wrought (Gothic bauht, brâht, maht, þâht, waurht); in others it appears in West Germanic, as sought (Gothic sôkid, Old Saxon and Old English sôht); in others only in Old English as laught (læht), taught (tæht, taht). But in the majority of cases the t is of later appearance, arising from the reduction of -ed to -'d, -d in Middle or Modern English, with consequent devocalization of d, not only after breath consonants, as in dropt, nipt, crept, slept, swept, left, lost, tost, past, but, in certain cases, after liquids and nasals, as in felt, spelt, spilt, dreamt, burnt, meant, pent; also in contracted formations, such as built, bent, lent, sent, spent, girt, cast. But in many words where the pronunciation has t, the current spelling is -ed, e.g. blessed, dropped, hushed, passed for blest, dropt, husht, past.
OED has a separate entry for the suffix -t as a formative of the past tense of some weak verbs and adds that: "In modern English on the contrary the spelling in t is more frequent in the past participle, esp. when used adjectivally, than in the past tense: cf. tempest-tost (see tempest-tossed adj.), the wind tossed the ship; in time past, he passed his time."
Additionally, it is not only used in British English. For example, shortening of a long vowel in the participle of certain verbs, as in crept, slept, the spelling with -t is universal.