The Nissan A engine is most definitely derived from the Austin B engine.

Nissan produced 20,855 Austin A40s and A50s from 1953-1959, the latter with 1500cc (1489cc) engine. They hired Donald Stone to design a smaller engine for the Datsun line. He destroked the BMC B series 1500 engine, and the resulting C1 engine was called the "Stone engine". This was fitted to the 1957 Datsun 110. When it was later increased to 1.2 liters for the first Bluebird (Datsun 310) it was named the E1. This was followed by the J-series engine, distinctly different from the Austins, but similar in many ways. Then the Nissan A-series for 1966 was the further evolution, almost completely different from the Austin in every way except basic dimensions. Cam is on other side, head is aluminum, etc. The 1982 Nissan E-engine was a direct redesign of the Datsun A-engine, hence the head bolts on, although it is OHC, lighter block casting, and shorter overall for easier sideways installation (FWD).

So, no -- Datsun cars never used Austin engines. But since 1957 they used new & improved Nissan engines based on the Austin design.