According to Pipephil,
"Conway pipes were most probably made by Singleton & Cole. If the pipe is stamped with a shape number this probability almost becomes a certitude. According to Wilczak & Colwell (op. cit.) Loewe had a line named Conway."
The Loewe Shape Chart lists Conway.
If you have any additional information about Conway, or who made them, please add it here, or send it to firstname.lastname@example.org and we can add it for you.