Your question is closely related to the academic research field of linguistic typology, which studies the similarities and differences of languages regardless of their actual relationship. There are many ways to look for an answer.
Comparative linguistics studies the relationships between languages. Since there are no proven (or perhaps even provable) relationship between the main language families, we could say that all languages which don’t belong to the Indo-European family are equally distant from all I-E languages like English, and this would of course include the vast majority of world languages. But obviously there are similarities that arose through contact between related or unrelated languages. These are usually called areal features. Where there is a lot of contact between languages over a long time, not only vocabulary but also significant grammatical features can be borrowed, resulting to a language area (Sprachbund), where a collection of common features are shared between a number of not necessarily related languages. This means that both I-E and non-I-E languages in Europe have similarities originating from Latin, Greek and French (so called Standard Average European), while both I-E and non-I-E languages in India have certain shared features mostly from Sanskrit. But apart from such similarities, there are a lot of features that appear independently in unrelated languages without a contact, and still are not different. These are compared by typology.
There used to be hopes in linguistics that a holistic typology would be possible; that is, languages could be compared as a whole, and that there could be an objective measure of how different one language is from any other. Holistic typology, if applied to every known existing language, would neatly answer your question. But unfortunately we don’t seem to possess such tool, so languages are compared by partial typology, where at one time just a single feature is being considered.
Many categories of partial typology exist: There is the word formation continuum ranging from analytic to synthetic languages, where English is somewhere in the middle and most different languages from it can be found on either extreme. The basic word order, where languages are categorized according to the order of verb, subject and object in a sentence, is another extremely common typology. English as subject-verb-object type is similar to other SVO languages, and different from SOV and VSO languages as well as from the three other very rare types. Typologies based on grammatic rules and sentence structures are both quite popular since they usually have predictive power: for example, most SVO languages tend to have some other shared features, too. Phonological typologies based on sound differences seem to be more difficult. Of these, the distinction between tonal and non-tonal languages, and the size of the sound inventory in a given language are particularly important. Here English is clearly a non-tonal language, and its number of phonemes is a bit above average, quite far from either extreme.
In my opinion, an actual scientific study trying to determine how different a given language is from English would need to make a large number of comparisons throughout the many categories of partial typology. This would have to be repeated for many potential candidate languages to see which one is the most different. Earlier posts give some good suggestions: Sentinelese we cannot currently study, though it’s probably related to the nearby Andamanese languages with unknown similarities. Piraha is a weird case. I’m not a linguist so I really can’t say anything about it, but it certainly is a controversial topic (there are multiple threads in SDMB about this). If Everett’s reports and interpretations are correct, Piraha could be fundamentally different from all other human languages, although how it could become so is an open question. Khoisan languages are also quite unique in many ways, and their “click” sounds certainly an interesting feature; however the clicks have been borrowed to some nearby Bantu languages, so they are not a fundamentally different feature.
One way to start typological comparisons yourself is via World Atlas of Language Structures, which provides maps of structural properties in world’s languages. It’s of course restricted a bit since not all properties have been studied in every language and so a given property will usually be compared only in a subset of all languages, but it’s definitely informative.
And, of course all this only applies to existing and well-known natural spoken human languages. The various sign languages all have their own grammars that also could be compared, and there are probably some contructed languages that are purposefully built to be as different from a typical European language as possible.