If you made the payment on time and coverage was supposed to start in June, the odds are everything will be retroactive. Save your receipts, you would be reimbursed.
What should you do? I've never dealt with COBRA. Do you make the payment to the insurance company or your former employment. If it is to your former employment, they are the ones you need to contact.
Getting a lawyer involved would probably be a waste of money though. Just keep bugging them yourself.