Legally, how much time am I allowed to have before I absolutely need to start a new job?

Dealing with an extremely disorganized HR. I won’t notify my current employer until everything (background check, drug test, and other steps) has been finalized. However, the company I am about to join is in the midst of changing their onboarding process and I haven’t even heard back from them after reaching out to them about starting the new position a bit later than the anticipated start date on the offer letter (As I would want to give my current employer enough notice after all the onboarding steps (background check, etc) have been completed. 
 

As a result, I am getting concerned that I may not have enough time to even give my current employer enough notice. The lack of responsiveness from the HR team at the company I am about to join has been frustrating to say the least. Would I legally be entitled to have a certain amount of time (like 2 weeks or so) after all the onboarding steps have been completed before I am absolutely required to start the new position?