Cavity wall insulation and building regulations by UK nation
In England and Wales, filling an existing cavity is notifiable building work: a building notice or competent-person self-certification is needed. In Scotland it does not need a building warrant but must meet the building standards. In Northern Ireland a building control application is needed before work.

England
The Planning Portal describes cavity wall insulation as specifically defined notifiable building work: a building notice goes to a building control body, which most councils do not charge for. CIGA-registered installers will in most cases submit it or self-certify through a competent person scheme. The material must be suitable for the wall.
Wales
The Building Regulations as they apply in Wales also treat cavity fill as notifiable work. Confirm the route on your quote, because Welsh policy can differ from England's.
Scotland
Schedule 3 lists work associated with thermal insulating material within a wall as not needing a warrant, on condition that the work meets the building standards.
Northern Ireland
Building Control NI's FAQ says a building control application is needed to insert cavity wall insulation into an existing wall, and Belfast City Council lists cavity wall and roof-space insulation as needing approval.
Related services
Sources: Planning Portal: cavity wall insulation and building regulations; CIGA: CWISC competent person scheme; Scottish Government: building work not requiring a warrant (schedule 3); Building Control NI: frequently asked questions; Belfast City Council: what are the building regulations.