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.