This feature is called Universal Links. You can read Apple's documentation on this here.
Basically for this to work the developer of the NYTimes website needs to have a file called
apple-app-site-association on their server. You can view that file on the NYTimes website here http://nytimes.com/apple-app-site-association. Within the file they can specify which URLs are redirected (on the NYTimes site, any that have a year between 2000 and 2099 immediately after the domain name) and which apps they are redirected to (four NYTimes apps). At a glance the file looks okay on their website, but interestingly it does mean that only links like this http://www.nytimes.com/2016/... will work, while anything else like this for example http://www.nytimes.com/reuters/2016/... will not. So right away it is worth noting that they've decided that they don't want all links to the nytimes.com website to open in their apps.
They also need to change a setting within their iOS apps to list which domains it is willing to receive links from. This takes the form of an entitlement called
com.apple.developer.associated-domains. Due to the way their app is configured I'm unable to check the plist files within their app to see if they are configured correctly. But in my own testing, this feature seems to work okay. When I go to this article I've randomly picked on DaringFireball and click on the NYTimes link, the NYTimes app automatically launches and opens the article rather than Safari.
What may have happened is that you can tell iOS that you don't want it to pass Universal Links from Safari to the relevant app. You do this by clicking on the link that appears in the top-right corner of the screen when iOS has opened an app with a Universal Link. From that point on iOS will ignore the Universal Link feature for that site. This is fairly non-intuitive, so you might have done this without realising that it disables the Universal Links feature for the site.
There are two methods to re-enable Universal Links for that site. Either long press on a NYTimes link and select "Open in "NYTimes" from the menu, or open the NYTimes website in Safari and click on the "OPEN" link in the Smart App Banner at the top of the site.
Let us know how you get on.