Is there an etymological connection between 輪{リン} as in 車輪{しゃりん} and "ring" in English? Or is this a false cognate?

There are a few things we have to look at to answer this.

Derivation of different Japanese readings

As we can see in the Jisho.org entry, rin is an on'yomi for the kanji 輪. On'yomi are the "sound readings", the literal meaning of the spelling 音【おん】読【よ】み, in reference to the sound of the kanji as borrowed from Chinese. So any question about the on'yomi of a kanji is ultimately a question about Chinese.

Middle Chinese pronunciations

Looking at the Wiktionary entry for Chinese 輪, and particularly the Pronunciation section, we see that modern Mandarin pronounces this as lún. That's not very close to Japanese rin.

⇒ Japanese on'yomi readings ultimately derive from Middle Chinese. Sometimes we know that a particular on'yomi is more specifically a 唐音【とうおん】 or "T'ang sound" borrowed during the T'ang Dynasty, or a 宋音【そうおん】 or "Song sound" borrowed during the Song Dynasty. At any rate, we can (mostly) ignore the modern Chinese readings. (Sometimes the modern varieties are informative for hinting at where a particular vowel value may have appeared. But that's beyond the scope of this thread.)

The Middle Chinese pronunciation is reconstructed as //liuɪn//. And if we click that ▼ downward pointing triangle to expand the section, we see a sampling of the different reconstructions from various academic authors.

I believe that part of the reason for the wide variety of reconstructions is that Chinese itself has so many different varieties. Think about English even, where different regions or dialects have different phonetic realizations of words. Just the first-person pronoun "I" can have several actual pronunciations, such as:

//aɪ//

//aː//

//ɑe//

Looking back at the Middle Chinese reconstructions for 輪, we see that some of them are even closer to Japanese rin than the "main" Zhengzhang Shangfang's reconstructed form of //liuɪn//, such as Pan Yuwun's //lʷin//.

So Middle Chinese //lʷin// (or something like that) was borrowed into Japanese as the on'yomi of rin.

Where the Middle Chinese came from

But we still haven't answered your initial question. :)

The Middle Chinese came in turn from Old Chinese, with a Zhengzhang reconstructed reading of //*run//. The character 輪 itself was composed as semantic (meaning) radical 車 plus 侖 as the phonetic (sound) base. We don't have any further etymological information about this (or at least, I can't find any based on sources to hand).

To rephrase your initial question, is Old Chinese //*run// somehow related to modern English ring?

Following up on the Chinese term, our data was inconclusive.

Where the English ring came from

Turning to the Wiktionary entry for ring, we see that English ring is traced back to Proto-Germanic hringaz ("ring; circle; curve"), derived from Proto-Indo-European (PIE) *(s)krengʰ-, derived in turn from PIE *(s)ker- ("to turn, to bend").

Comparing reconstructed ancient roots: Old Chinese and PIE

PIE *(s)krengʰ- and Old Chinese //*run// don't match up very well.

However, PIE is tentatively dated to around the time of the domestication of the horse, maybe 6000 years ago. Old Chinese is reconstructed to around 3270 years ago. That's a lot of time -- so we must still ask if the PIE *(s)krengʰ- might have given rise to Old Chinese //*run//.

We know from long research into sound correspondences between PIE daughter languages that the change from PIE //(s)kr-// to Proto-Germanic //hr-// was a regular shift. We can also see from the list of descendants that all other branches of PIE include that hard //k// at the start, with or without the //s//. So the only branch where the initial //(s)kr-// in reconstructed PIE *(s)krengʰ- lenited (softened) into just //r-//, as in both modern English and Old Chinese, is the Germanic branch -- the branch of PIE that was most distant from Old Chinese speakers.

Conclusion -- tentative

Given the state of current academia (as publicly accessible by me at the moment, anyway), it appears that English ring and Japanese rin are only accidentally similar in sound and meaning.