When Should You Disclose Availability/Unavailability to an Employer?

I am currently studying abroad in London. I am now interviewing for a position (whose start date I now realize was specified as May 31st) and don’t know when to tell them when I’ll be back. I dont yet know when my last exam is, so there’s a chance I’ll be back in time. But, the absolute last day of all exams is June 8th so I may not be back until then. Do I tell them during my next interview? After getting the job?

I fear that if I say anything I will ruin my chances. But, I also fear that if I don’t say anything and do get the job, they’ll think I purposely ignored the start date and lied to them. I just hope they’ll be willing to work with me

I would mention this after receiving an offer.

It’s only 8 days - if it’s a FT position, that’s nothing.
Unless the job requires leaving for a new planet and the rocket takes off 05/31 this shouldn’t be an issue. If it’s a summer internship, however, it might be a problem because some companies train all their interns at once.