Au contraire: the very few examples of London Mint "no H" 1882 pennies surely proves that only one die set was used. Moreover, the examples extant could quite possibly have been struck in order to test the new electronic minting apparatus; given that the entire mintage for pennies had been 'farmed out' to Heatons, there would have to be a special reason for striking a few pennies at London. Testing the equipment seems as good a reason as any, and once struck, the few examples for testing would have been released into circulation as no-one would have thought then that the existence or otherwise of a mintmark would have been of interest. (!)
If a coin leaves the Mint with no H , then it is definitely a "no H " wherever struck. But given the very large number that appear on eBay (worn), it's clear they left the Mint WITH an H and it has been worn or filed away; however if such an example with the wrong dies is in high enough condition e.g. VF, one would then - in all reasonable suspicion - look to the expertly created 1933 penny fakes and strongly suspect that a machinist had been at it.
As for 1922 cents with no D caused by die fill, fetching large sums, I guess that indicates the gulf between US and UK collecting. We had a 1961 halfcrown variety with the designer initials "EF" missing on the reverse; initially this was thought to be a die error and examples fetched a pretty modest premium over book price. Once it was discovered that the 'error' was caused by die fill, interest fell away quite quickly, to the extent that you will find no reference to them now.