Why isn't it 3rd August?
Because it takes 21 days to switch supplier.
And how do I get energy supplied between the 3rd and the 21st?
As detailed on the moving out process that you followed:
If you haven’t already, you’ll need to let the current energy supplier know that you’ve recently moved in, so they can charge you for the energy you’ve used so far.
You’re not supposed to start the switch to Bulb for your new property until you have legal responsibility for that property, which is your case sounds like that doesn’t happen until 3rd August. What you’ve done is start a switch of a property that is owned/occupied by someone else, and they may well cancel the unexpected switch when they are notified.
Bulb really should make it more clear that when they say
If you want your new home to be supplied with renewable energy, then joining Bulb again is easy. Just head to the sign-up page and let us know your new details.
They mean give your new details AFTER you’ve moved in and they have actually become your new details, NOT give your soon-to-be new details.