Barry Jones recalls in his book 'The Meteor', that an 'exhaust reburning system' had been discussed by RAE before 1945. An experimental system was tested in Meteor EE215 in April 1945.
In a similar time scale Rolls Royce had been investigating an 'after burning' system, but fully operational systems for Derwent engines were not available until 1949; these had variable nozzles.
Of interest John Grieson's book 'Jet Flight', Jan 1945, did not mention reburning systems, even though advanced concepts of turboprops and bypass were foreseen. In a chapter by George Carter (designer of the E28 and Meteor) on future engine systems, he noted a divergence between civil and military designs, but this section was subject to security restrictions at the time.
From the above it might be concluded that 'reheat' preceded 'after burning'; perhaps the latter coming from a 'different' system designed the USA, or a corrupted amalgamation of the preceding terms.