Your UK issued JAA PPL is an ICAO PPL.
This should entitle you to noncommercial VFR flight anywhere in the world, in a plane whose registration country is the same as your license issue, i.e. G-reg.
If you want to go to Mongolia (let's assume they are an ICAO signatory) and you want to fly a MG-reg (if that's what it is
) plane over there, they might just let you, but more likely they will require you to go through some license conversion/validation process.
The above is all one can say, I think, in general terms.
It varies hugely from one place to another. For example, any ICAO license holder can fly a G-reg, worldwide, without any formality, because the UK ANO automatically validates any ICAO license for flight in a G-reg. But this is rare; in most places you have to go through some paperwork, as a minimum.
In your case I would contact the HK aviation ministry - if they have one.
Separately from the above, the place you rent a plane from is likely to have minimum-hours and/or a checkout flight requirements. This could be a problem for a new PPL holder.