Allenson, a native of Durham, matriculated as a sizar of Trinity College, Cambridge, in 1576; but in November of the same year he migrated to St. John's College, where he obtained a scholarship on Mr. Ashton's foundation, and became a pupil of the Puritan Dr. William Whitaker, whose religious principles he adopted.
On 20 March 1583–4 he was elected a fellow of St. John's College on the Lady Margaret's foundation.
Baptismo et Eucharistia, and in the dedication to Dr. Tobie Mathew, Archbishop of York, informed him that Dr. Whitaker had not himself published these lectures: quæ tamen de Sacramentis adversus Bellarminum in Scholis Academiæ publicis prælegit, vir diligentissimus D. Allensonius, collegii D. Joannis Evangelistæ socius, fideli calamo ex ore dictantis excepit et post authoris mortem, cum ipsius D. Whitakeri concisis annotiunculis in memoriæ subsidium scriptis, accurate contulit præloque destinabat.
Sed ex rerum humanarum vicissitudine, ipse etiam, antequam prælo mandarentur, fatis concessit, exemplar prælo destinatum post ejus obitum ad manus meas pervenit.
Thomas Baker, in his History of St. John's College,’ asserts that the life of Whitaker was written by Allenson; but this is certainly a mistake, as the author of the biography was Abdias Assheton.