Pushing back start date?

I got a job offer three weeks ago. At the interview they told me the hiring process will take 2-3 months. I was under the impression my start date won’t be until after that time period. It seems that the process has been going by fast than what was stated. The job is 40 min away from where I currently live. My plan was to get all of my things situated and move closer to the job by the end of May. In the case they give me a start date sooner than what’s good for me, how can I go about this with the company? Ideally I wanted to start in June. At the interview they also asked me if I have anything planned within the next few months, which I do. I told them I have something planned in May. Since they emphasized the process will take 2-3 months it slipped my mind to let them know I will be moving closer to the job. Anyways, any tips?