'It was a bitter night, so we drew on our ulsters and wrapped cravats about our throats. Outside, the stars were shining coldly in a cloudless sky, and the breath of the passers-by blew out into smoke like so many pistol shots. Our footfalls rang out crisply and loudly as we swung through the doctors’ quarter, Wimpole Street, Harley Street, and so through Wigmore Street into Oxford Street. In a quarter of an hour we were in Bloomsbury at the Alpha Inn, which is a small public-house at the corner of one of the streets which runs down into Holborn.'
It is quite impossible to trace completely the path Holmes and Watson took in BLUE when they left 221b and headed to the Alpha Inn to inquire about geese. As with every walk we take, we don't always follow the shortest route, but sometimes the most scenic or the most familiar.
But we can come close.
There are several different routes they could have rambled, all of which could touch upon the streets mentioned in the quote.
And we have to realize that some of the street names have changed or been rerouted.
The walk would have been right around two miles, so when Watson said, 'In a quarter of an hour. . .', he must have meant from when they hit Oxford.
If you do a google map route, it misses Oxford street all together, but does hit all the other streets.
However, if you were to turn south on Harley Street and connect with Holles St. you would hit Oxford. After Harley St. we could also turn south on several streets to get to Oxford.
After about Regent St., the further east you go on Wigmore the more your path would start veering away from the Alpha Inn.
And we must remember, the quote does not actually say they went to Holborn, just that the Alpha Inn sets on a corner of a street that runs into Holborn.
These would have been familiar streets to Holmes and Watson. They would have traveled them a lot.
It would be a wonderful walk to take sometime.