Residential housing take up only ca 5 percent of UK land. The only reason it's that high is due to huge amount of low density housing (e.g. my terraced road w/houses and gardens take up ca. 40,000m2, near my local station the same number of people will soon be housed on ca. 2,000m2). Space is not an issue. Other infrastructure, maybe, but UK is going to need increasing amounts of immigration over coming years to offeset below replacement fertility rates to prevent a demographic crisis.

With fertility rates plummeting worldwide, there's no realistic scenario where the UK will be overrun.