New answers tagged shared-hosting
A better way to to do this is to instead of a subdirectory, use a subdomain like blog.website.com. The reason for this is that a DNS record can be made to the other server for the subdomain, which is not possible with subdirectories. A subdirectory is possible, but you would have to use frames, a .htaccess redirect or another way which is totally not stable ...
Does it have to be website.com/blog? A faster solution and less complex would be to just host the blog as blog.website.com For website.com/blog, your best solution is to employ URL Redirects.. see: http://serverfault.com/questions/567476/permanently-redirect-subfolder-blog-to-subdomain-on-another-server
This clearly isn't a DNS problem since "wait time" is counted after domain was resolved and connection to server has been established. What you can start with on WordPress level is getting a plugin which will show you is this time is actually spent inside WP page load. Laps (disclosure — developed by me) or Query Monitor are what I look at first for issues ...
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