Primark interview

I’ve recently had an interview with primark at Oxford Street (Tottenham Court Road) and by the end, the interviewer offered me the job straight away but I haven’t heard anything from them yet as he said I would. Plus, during the interview he asked me if I had any holidays already booked, which I thought I didn’t, but my partner booked a surprise holiday for our anniversary (in a month) and I got to know today and, I thought I did not have to give a notice to my current job as I am in a 0-hour contract but I do, it’s only a week, but I did not inform Primark. Should I ring them and inform them of all of this? I’m scared I’ll lose the job offer.