Você tem 90 dias a partir da entrada no UK, definido pelo seu cartão de embarque. O texto oficial que responde a sua pergunta está neste documento oficial do governo do Reino Unido que detalha o CTA (Common Travel Area), no final da página 34 e início da página 35.
A entrada no UK pela Irlanda é chamada de "Deemed leave":
Deemed leave, as suggested, is leave that has not been endorsed by way of a stamp into an individual’s passport – as individuals travelling from Ireland to the UK will not necessarily encounter a Border Force Officer.
Where the deemed leave provision does apply, a person is usually treated as having 3 months leave to enter the UK, starting from the date they entered the UK evidenced by their ticket or boarding pass, with a prohibition on taking employment or occupation for reward. However, in some cases the person will only have 7 days to regularise their stay. The CTA entry through Ireland flow chart provides greater details of the deemed leave provisions.