Hi Big bear, it will depend on your payroll system and how late in the month they can add you to the system and organise pay for you. In our place payroll closes at the end of the second week in the month (for a 30th payday) so we would struggle to get pay to you on the 28th of June if you weren't starting until the 22nd.
However, you may be able to ask for it to be processed as an emergency payment? Or, their system may be flexible enough to be able to pay you for that first week at the end of June - perhaps give them a call to check.
Is there any kind of hardship loan from the job centre that you can apply for?
Sorry for so many disjointed thoughts there!!! Good luck with new job xxx