Banks that deal with the general public,' High Street banks' whether they are foreign owned or not, are incorporated under and are bound by English law. The local version is a subsidiary company but legally English So, to that extent they are all the same. What used to be 'the Midland' is now the Hong Kong and Shanghai Banking Corporation (HSBC) but is just as British as the Midland Bank was in the old days.
If you are thinking of having a bank account abroad, that bank abroad , whether it be British owned or not, is governed by the banking laws, practices and codes of the country it is in. So Barclays in France looks outwardly like any Barclays here and is, indeed, part of Barclays Bank's group, but it's incorporated in France and all its practices and any codes and laws that apply to it are French practices, codes and laws not British ones.That applies to any 'guarantees' that may exist.
I can't think why you'd be worried. There's no comparison between the big banks abroad and a tiny (in finance terms) business like Northern Rock. Societe Generale took a hit of billions and that hardly registerer with them !