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@The bullies win wrote:

How do you handle the procrastinators who expected you to file the extension in April, but failed to contact you? 

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You can't file an extension for their 1040 unless you are able to properly estimate their tax.  That means unless you have most of their information, you don't file an extension.  But you could send out emails in early April telling them to make an electronic extension payment for the tax return (even if it is only $1) and that the IRS claims they will grant an extension based on that payment.

As for the deadline, YOU need to set YOUR deadline, and stick to it.  IF you are willing to accept things after your deadline, make the client aware that there will be a very large additional fee.