'collectively-led' is a modern compound formed from 'collectively' + 'led'. 'Collectively' derives from Latin via French and English: 'collectivus' (Latin) > 'collective' (French/Medieval Latin) meaning 'gathered together'; 'led' is the past/past-participle form of English verb 'lead', which goes back to Old English.

'collectively' evolved from Latin 'collectivus' (from 'colligere', 'col-' + 'legere' meaning 'to gather together') through Middle French/Medieval Latin into English as 'collective' and then 'collectively'. 'Led' comes from Old English 'lǣdan' (to lead); its past form developed into modern English 'led'. The compound 'collectively-led' is a recent English formation using the adverb + past participle pattern to form an adjectival phrase.

Initially the parts meant 'gathered together' (collect-) and 'to guide or cause to go' (lead); over time they combined in modern English to express the idea 'guided or managed by a group', which is the current meaning of 'collectively-led'.