Both are correct, but they have different meanings.
- If you want to say that he is always on time
of those men functions as an attributive adjective, qualifying 'one'. To understand how you should phrase a sentence, it can help to remove extra elements like this, which could be confusing. When you take this out, you have "He is the only one who [is/are] always on time". Because 'who' is the subject of that verb in the second clause, the verb must be 'is'.
- If you want to say that they are always on time
Consider the phrase they are always on time. If you already had a sentence like 'they [the men] are here', you could add this information as the men, who are always on time, are here. In your sentence, the second option matches this structure - he is the only one of those men, who are always on time.
This makes sense except that when you take out the descriptive clause and adjectival phrase, you realise the sentence you have says that 'he is the only one', which is grammatically correct but not really semantically correct because 'one' is not defined - a reader would tend to ask 'the one what?' and because of this, the reader expects a 'who is [something]' phrase about the man. Because of this, the reader will typically assume your 'who' phrase to have the verb 'is' to apply to the man, and would assume that 'are' is just a mistake. To prevent the reader from assuming and then asserting that you are wrong, insert a comma to distinguish that it is only a relative clause and not integral to the meaning of the sentence: "He is the only one of those men, who are always on time".