The reason to move over to SSL before October is so that when customers go to your client's contact page to call or inquire about a service it doesn't show them that the website is not trusted. Which would make them think the business isn't trustworthy and cause the business to lose a sale.
And yes, it will help SEO too

also potentially conversions. SSL is just great to have these days for a whole host of reasons. Http is a thing of the past, this change is just ushering it further in.
I've done a lot of SSL migrations (for local businesses, smaller sites) and it isn't too daunting.
The redirect is simple. I just do it in the .htaccess folder. Here's the code I use:
RewriteEngine On
RewriteCond %{HTTPS} off
RewriteRule ^(.*)$ https://%{HTTP_HOST}%{REQUEST_URI} [L,R=301]
It will redirect all pages to https. Just make sure you site URL structure remains the same for all pages. I don't know why you would change the site URL structure during an https migration anyway though.
The main issue you'll probably have is securing all the http elements on the website after the fact. Which just means changing the call from http to https in the code, like images, etc. If not all pages are secure it will still throw insecure warnings even if you've installed https.
If you're on WordPress securing http elements is pretty simple to handle. You'll just need to change the site in settings > general to the https version. That should take care of most of the issues.
Use whynodpadlock.com to figure out the rest.
Good luck!