The best answer to this question will really depend on you. Are you working with an experienced loan officer who understands the FHA loan process? Do you have all your required paperwork ready to submit?
Is the mortgage company or bank FHA direct endorsed? All of these items will determine how long it takes for your transaction to occur.
I have over 18 years experience in the business and first time homebuyers are my speciality. I write almost 95% FHA mortgages. Feel free to contact me. I would be happy to help you.
It's sounds like you are in good shape and should have no worries whatsoever. You loan was only taken 6 days ago, I am not sure if you are a first time homebuyer or not but mortgages usually take a little longer than your typical credit card, personal loan, or car loan. I think you will be fine an you probably will actually close before the 15th of July. But to answer you question, it really depends on your scenario...in your case it can be easily done in 15 days!