Having read what a gestoría is, and what a 'gestor' does I can say that we don't have anything equivalent in the UK. People sort our their own documentation directly for things like passports, car insurance/tax and work permits (an employer might have someone in-house to help with a work permit but you wouldn't go to a specialised agency.) Many people use accountants to give financial advice and help with tax/VAT returns if they are necessary, for example if someone has their own business or a high net worth. And for anything legal, people would consult with a solicitor and then seek out more specialised advice if necessary.

I looked up a translation of gestoria and a description of what it does and it is hard to see anything directly similar in most English speaking countries. http://dictionary.reverso.net/spanish-english/gestor%C3%ADa In most English speaking counties you would approach be expected to go to each government agency separately and transact your business in person or on line. Some people do use agents to prepare their income and other tax papers for filing but that is about it. In most English speaking countries governments would be criticised if citizens were unable to do this business without employing a third party.

An accountant or bookkeeper would deal with tax and financial affairs, but would not normally deal with other business matters.

Possibly a bureau? I guess there isn't a common word, as it isn't a common business - here at least.