Yes, the foundations of the constitutions are not the same, one of them has a mostly codified constitution, the other has a mostly uncodified (uncodified but mostly written down, that is not a contradiction!) constitution. They both have constitutions however, so the phase "Britain has no constitution" is wrong. To be clear, I am not saying that is good or bad that Britain has an uncodified constitution, just that from my definition (and most political and legal definitions) of what a constitution is the phrase "Britain has no constitution" is wrong. Britain of course has laws, and laws about how new laws are made, etcetera. This forms a constitution.