You could go through a solicitor and change your name by deed poll (from 1914 these are also shown in the London Gazette, I think the NA or someone has some earlier records) - but unless there was some good reason to have the details legally noted down, and you had the money (so, upper class, where the name change might effect or was due to inheritance issues, perhaps), in the 1800s this was not all that common to my knowledge. You see a lot of people listed as
alias a lot in records, e.g. from the Old Bailey:
multiple alternate names:
Thomazine Davies, alias Sparks, alias Bailey
Mary Williams, alias Smith, alias Jones
James Harris, alias Levingstone, alias Barrow
minor changes:
Francis Pevanson, alias Peverson , a French man
Jane Stanley, alias Stavely
John Collet alias Cole
nicknames also:
Catherine Cotterel alias Irish Kate alias Catherine Irise alias Twford
John Bennet alias Freeman , but more notoriously known by the Name of the GOLDEN FARMER
Later on you see quite a lot of them in the London Gazette for immigrants (particularly post-WWI switching from German-sounding names, for example). Or if they naturalized then both names might be listed:
i.e. from the NA:
Naturalisation Papers: Bentel, Adolphe (also known as Charles Boydel), from Frankfort-on-Mayne.
I would suggest that you look into the Savage family background and keep an eye out for anyone surnamed Thomas. Do you know their occupations, and did all the family move to London? Does anybody switch back and forth between the surnames?