When receipt duty was abolished on 31st January 1971, firms and organisations were allowed to use their remaining stock of overprinted stamps for a period of six months until 31st July 1971.
It is therefore not surprising to come across overprinted stamps on cover used during this limited period. However, one is just as likely to find overprinted stamps on cover outside of the legally permitted time period.
The earliest such cover from my own collection is dated 13th June 1882, with a 1d lilac overprinted by the Croydon Local Board of Health. This seems to have gone through the postal system without any problems and postage due was not charged.
The latest example of a postally used overprint I have seen is on a cover dated 15th September 1976 which has four 2d light brown Wilding stamps with N T G B overprints. Postage due of 13p was charged.