I live in Australia on a long stay Temporary Residents visa and do not require to declare overseas income to the Australian authorities that is taxed in UK.
I get the benefit of the UK tax free allowance applied to UK earnings and the Australian tax free allowance on my Australian income. It may not pay to go for the tax-free in UK option if the tax payable in, say, NZ is going to be higher than splitting tax paid between the two countries.
In several countries, Australia included, the UK pension stays frozen at the date it was first drawn or you leave the UK, whichever is the later.