I applied last year September to work at Heathrow but only received the formal job offer this May 2023. So nearly 8 months, but thankfully they scheduled my start date according to my availability/holidays, giving me a month to work my notice period. As long as you inform them of your notice period when they contact you to offer you the role, from what I have heard, they can work around it. If you’ve already been offered the role, I would think probably 2-3 months to have your vetting all sorted. Some people get it faster, others like me it was months later , but hold on on giving your notice until you actually formally accept a starting date with the signed contract.