The distinctions between IPA pairs such as [o]~[ɔ] and [ɔ]~[ɒ] are not obvious
In the context of the transcriptions for the word coal, the difference between the transcription /ɔʊ/ (for British English) and /oʊ/ (for American English) is not meaningful on the phonetic level. The International Phonetic Alphabet (IPA) symbols /ɔ/ and /o/ represent adjacent vowels.
Theoretically, IPA [ɔ] represents a "mid-open" back rounded vowel, and IPA [o] represents a "mid close" back rounded vowel.
But in actual use, the IPA is vague enough that the qualities of vowels that are transcribed /ɔ/ and /o/ often overlap. So the use of the symbol /ɔ/ vs /o/ is commonly a matter of arbitrary convention, sometimes based the history of a sound or on its value in a particular dialect of a language. If you see a single transcription that uses both symbols (such as "/otɔ/"), you can conclude that the symbols "o" and "ɔ" are meant to contrast, but otherwise, you should think of them as potentially representing the same vowel quality. So far, I haven't seen a transcription of any accent of English that makes a contrast between /ɔʊ/ and /oʊ/.
The distinction between [ɔ] and [ɒ] is similarly vague. Michaelyus posted a comment linking to a blog post by the linguist John Wells that uses a third transcription for the vowel sound found in coal: [ɒʊ].
Even if there is a difference between the two accents in the pronunciation of words like coal, the closeness of the different vowels involved would probably make it difficult for you to hear the difference.
The vowel in "road"
The difference between the final consonants of coal and road is relevant,
because for many speakers the word-final L sound in coal causes the preceding "long o" vowel to be pronounced differently from the way it is pronounced elsewhere. That is why Wiktionary provides a "UK"* transcription with /ɔʊ/ for coal, but not for road.
*As user6726's answer says, the labels chosen for these kinds of transcriptions can be problematic since the transcriptions don't in fact represent "British" or "American" pronunciation as a whole. The label "Received Pronunciation" or "RP" avoids that potential false impression—it's pretty clear that not all British English speakers have an "RP" accent—but runs into the issue that people don't agree about who does have an "RP" accent.