What is the consonant equivalent of Well's lexical sets for English vowels?

One of the benefits of lexical sets is that they make communication possible across dialects and IPA conventions without positing a single reference dialect which would also have to include an idealised speaker of that dialect.

If you say, dialect X has pronunciation Y for vowel X, you may not know what you're talking about because of the way IPA is used as well as the great variation across dialects.The lexical sets also don't just correspond to individual sounds but rather to collections of words that have a particular sound which can generally be identified. The distribution of the vowels is also not uniform. So you cannot say, in dialect X, replace ɑː with æ but you can say that BATH and TRAP vowels have merged. Or you could say, that some speakers pronounce the i in the HAPPY lexical set as schwa without saying anything about the pronunciation of i in general.

None of these things are an issue with consonants. The IPA is consistent across all dialects, the variation is less and the distribution is more uniform. So you can say things like in context Z, consonant X is pronounced Y. (E.g. intervocalically, voiceless stops are pronounced as glottal stops). Having a lexical set here would only confuse matters.

It seems that what you want is simply examples of consonants in English. And those, of course, are readily available.