[Julie Steiner]

Moot now, since you have gracefully sidestepped the issue in the current revision, Tony, but here's my take:

The short version: What Rogerbob said.

The long version:

The pronoun "who" is the subject of the verb of its own clause, so it's doing its own thing grammatically and should be nominative, regardless of the case of the personal pronouns it modifies. And those personal pronouns are also doing their own thing grammatically, regardless of whether the clause that modifies them begins with "who" or "whom."

One wouldn't say "me, who walked"?

Actually, one might.

At the beginning of a sentence of which the first person is the subject of the verb, you are correct that one would say something like "I, who walked everywhere, arrived late to the party."

But at the end of a prepositional phrase in which the first person is the object, one would say something like "Others would arrive late, like me, who walked everywhere; but they generally had flimsier excuses."

"Like" is often a preposition, in which case its objects would be "you and me." "Like" can also be a conjunction with an omitted but implied verb, as in "like you and I [are]" or "like you and I [do]." (I'm old enough to have been told to use "as" or "such as" instead of "like" when a verb, or the ghost of one, was involved, though nowadays "like" can be used in those situations, too. But that's not what was going on in this situation, anyway.