How do I start from scratch in the middle of the month on a biweekly budget with both monthly and biweekly bills

Should I just wait until June first or will I be messed up? I’m frustrated with thinking I’ll have to input a whole months worth of data (no cushion yet) for it to make sense?

Do you have access to your balance and cleared transactions on line or in a banking app? If so, you can start with the current balance, add any expenses you’ve paid but which aren’t reflected at the bank yet, prorate your envelopes for the remainder of the month/fortnight, and have everything reset at the start of the next budget period.
Your budget will likely change quite a bit for the first several months as you get a handle on your spending; just be sure the “must pay” envelopes (utilities, rent, groceries) are filled and leave a bit of buffer for the less critical areas (clothes, dining out, entertainment). Whether you set things up monthly or biweekly depends on your preference, but if you don’t get paid biweekly then a monthly budget is likely easier to track.