No. I do not believe that would effect processing. They use the first four letters of your last name and the last four digits of your ssn as an identity confirmation for e-file.
Once your return is filed, TurboTax has no control over the processing of your return or the release of your refund.
Here are a couple FAQs to track your refund:
When will I get my federal tax refund?
How do I track my state refund?
Why does Where's My Refund say "approved" but TurboTax still says "accepted"?
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