Hi Simon, welcome to WikiWirral.
Sorry to say, but your guess at the Pub
on the corner of Getley St is wrong. That would be the Manor Arms at 180-182 Chester St, directly opposite the end of Abbey Street. On some maps it may say Mersey Place, but that was rebuilt and became Gatley St between 1899 and 1911 according to the old maps.
The Dutchess of Edinburgh seems to have been about six doors down from the Manor Arms, on the same side of Chester St. If you look on the 1964 map I posted, it would be about where the letter R appears in the 'ChesteR
N.B. The Manor Arms mentioned above is not to be confused with the Manor Arms /Manor Hotel at 9 Union St, which is less than a mile away.
There are some ariel photo's of the area which were posted recently, under a different topic, from which you might be able to locate the Dutchess. If I find them first, I'll append them.
Good luck with your search, and if you do find a photo and more info, please share it with us.