I'm not a fan of auto detection and redirection schemes. I avoid sites that won't let me go to the site I choose.

Stick with something simple, like small flag icons the user can click on to go to the desired site.

I like the way, eBay does this. It will let you proceed to your local site but you can also choose to keep browsing where you are

in general, the best way to do this is to offer the initially requested .tld site as a default.
then maintain the user's most recent .tld preference or selection using the usual methods.
if you have a login site or are otherwise maintaining a session, you can keep the .tld selection in the session or user data.
otherwise you can keep the .tld selection in a cookie or in the url.
once you have a detected a preference that doesn't match the current .tld, you respond with a redirect to the preferred .tld.
that redirect response also needs to include a cookie with the new .tld preference.

you can detect what is happening with a site that has the behavior you desire.
see what responses and cookies you get as you select a language/country/flag or login and then see what response you get when you return.