The English version of a name often originates from what a record keeping scribe thinks that they have heard when the name is pronounced. It is then translated into whatever English name the particular scribe is familiar with. This can lead to a multiplicity of versions and hence the important assistance that can be offered by soundex search systems.
In England however Italian names are frequently retained in their original form because the italian pronunciation is quite clear, distinctive, and recognisable in the English language. For this reason you are unlikely to find English versions of Jaconelli which are English corruptions of the original.
If the name has been Anglicised the new version will have been a deliberate choice on the part of the name holder. It may bear no soundex relationship at all to the original. They may choose Jackson to retain some feel for origins but equally may choose Smith if they wanted to blend into English culture.
The answer has to be that we simply do not know, but name changes may be recorded by deed poll if it occours in recent times. Check out this link if you would like to know more about deed polls: www.adviceguide.org.uk/england/relationships_e/relationships_birth_certificates_and_changing_your_name/change_of_name.htm.