/u/hlly is right. I'm going to give you the same answer but I'll expand on it a little bit.

The tl;dr is that English isn't special.

It would make sense to me that English might have changed more than other languages due to various cultures ruling what is now England…

To give an analogy, China has been ruled not just by Chinese but also by Mongols, Jurchens, Manchus (a later version of the Jurchens you could say), and in some parts by the Portuguese and British, and in Taiwan's case, by the Spanish and the Dutch. Add to that the hundreds of other indigenous groups which have had varying degrees of impact on the language, and what you find is that Mandarin (and the earlier stages of what became Mandarin) has changed massively. Lots of countries have been ruled in whole or in part by other cultures or people with other linguistic backgrounds. Even those which have not had this happen in the recent past, you still have people speaking different dialects taking turns being in charge and imposing their form as the standard, which then trickles down to others.

as well as the language having been spread globally to some extent for centuries by now.

There's a small issue with this idea, which is that you're then looking at English as "All Of What We Currently Call English", but you'll notice above that I say Mandarin instead of Chinese. In reality, even just what we call Mandarin in the most general sense might actually contain things that should reasonably be considered not Mandarin at all, but a separate branch, a sister language to Mandarin. There's a language group called Jin spoken in Shaanxi province that linguists have debated about fiercely since at least 1985 (thanks a lot Li Rong) as to whether it's Mandarin by descent or just seems that way because of exposure. There are good arguments on both sides of the debate.

Why is this a small issue, or an issue at all? You can say something like Variety X has changed a lot compared to its parent form in regards to things like pronunciation, word order, word choices (using 'radical' to mean cool, if radical has caught on more). You can say that Variety Y is more conservative than Variety X in terms of pronunciation, and so in that regard Variety X has changed more. But what you're asking about with this "global" word isn't so much the change as it is the divergence.

In which case the answer to the question "have other languages diverged as much as English" is a bit harder to justify as "yes", unless we're also looking at globally distributed languages like French, Spanish, Portuguese. Languages of colonisation. Macau (China, next to Hong Kong) has a Chinese/Portuguese creole. That's pretty different from what someone sounds like in Lisboa.

But then you can still count Chinese languages among those, thanks to diaspora communities. Cantonese in Hong Kong sounds quite different from the variety spoken in Kolkata (Calcutta) and different still from that in Washington DC. These are all fairly recent divergences as well, much like English.

I suppose someone could be considered fluent in English and yet not be able to completely understand a Shakespearean play, is this in any way unique?

Nope. The average person doesn't understand well the things that the culture places as the height of literature. Literary registers are always something requiring some education in. You can't just go pick up the Arabic or Spanish equivalent and be down to party without first needing some formal training on it.

Are there other languages where the "line" that clearly represents the same language but is largely unfamiliar to a native speaker is noticeably further back in time or closer to the present day?

Part of the issue here is also one of terminology. We call Old English "Old English" but maybe we could call it Old Scots instead and be just as accurate. Then would that change an English speakers idea that it's the same language?

Then part of the issue is, "same language" by whose standards? This ties into the terminology point. I can tell you that to a linguist, modern English and Old English are most certainly not treated as the same language. Grammar is different, vocabulary is different, pronunciation is different.

However no, there is never any magic line in the sand. It's all a matter of degrees.

I'm not 100% sure on if you mean "line" as lineage or "line" as line in the sand which one crosses. If you can clarify, I can expand on that. But otherwise if you mean what I think you mean, then no.

This is a question of historical linguistics and would best be posted in /r/linguistics. The answer, by the way, is yes; English is not as special or unique as many of its speakers believe. Many languages have undergone much more extreme change. Dyirbal, for example, has its 'line' very close to the present day, while Icelandic has it much further back than English (note that I don't really subscribe to the idea of a 'line' because language change is fluid).