World aviation is regulated by the ICAO
Not true. ICAO produces Standards and Recommended Practices from which individual member states develop legislation. EASA is in a rather strange position, being neither an ICAO member state nor a regulator but simply a rule maker.
Post Brexit, a UK-issued licence will be valid to fly a UK registered aircraft anywhere in the EU in exactly the same way that, for example, an FAA licence is valid to fly a US registered aircraft. Since a UK-based operator will not be "established or residing in the Community", there will no requirement to hold an EASA licence. The difference is that a UK-issued licence will not be valid to fly non-UK registered aircraft.
It is also worth considering that a UK-issued LAPL, being a sub-ICAO licence, may not be valid outside the UK after Brexit. The main reason to hold an LAPL is the less stringent medical requirements and, given the self-declaration process for a UK PPL medical combined with the inevitable removal of the distinction between EASA and Annex II aircraft, we will probably see the LAPL disappear.