The existence of a 1920 penny with obverse 3 is a little difficult to explain – there is apparently only one example known and it currently resides in the British Museum (p212, Freeman, The Bronze Coinage of Great Britain, 2006). Possibly it was produced to test the suitability of the newer obverse 3 but given the eventual adoption of obverse 3 for a majority of 1921 and 1922 it seems unlikely that just one example would have survived, and it also seems unlikely that a pattern for something as mundane a new obverse die to reduce ghosting would be sent to the British Museum.
The existence of 1921 pennies with obverse 3 on the other hand is easily explained by the introduction of yet another new obverse die to combat ghosting part-way through the production of 1921 pennies – V.R. Court's survey of pennies (p67, Coin Monthly, September 1972) shows that around 37% of 1921-dated pennies have obverse 2 and about 67% of 1921-dated pennies have obverse 3, so obverse 3 was either created part-way through production of 1921-dated pennies or it was created at or before the start of production of 1921-dated pennies and the Royal Mint chose not to waste their remaining stock of obverse 2 dies.
A possible explanation for the existence of a 1920 penny with obverse 3 is that the British Museum requested an example of a 1920 penny for its collection late in 1921 and the obverse die that was used to produce it just happened to be a newer obverse 3 rather than the expected obverse 2 which likely all other 1920-dated pennies were struck with. Certainly in 1908 and again in 1919 it appears that the British Museum had not kept their collection of British Indian coins up to date and requested that both the Bombay and Calcutta Mints send coins of every date for the British Museum collection (vii, Stevens & Weir, The Uniform Coinage of British India 1835 to 1947 A Catalogue and Pricelist, 2012), so there is a precedent for this.
The British Museum's registration for the 1920 obverse 3 penny is 1921,0326.11 which, while not entirely reliable, does suggest that the penny was acquired and hence likely struck in 1921.