Caslon 540 italic.
http://www.myfonts.com/search?search%5Btext%5D=caslon+540
Looks so weird all italic caps.

I don't find it weird, but the way the tail of the Q crashes into the line below I would think it isn't that great. But, I'd be curious to see more.

It is pretty awkward, isn't it? Rather unfortunate, given how well it looks otherwise....

Well, to me it looks like the typeface was never meant to be used like this (especially THIS typeface). That’s probably why it was chosen. And that makes it an interesting design. Very UK.

I think Caslon 540 it is...Here is another visual of the font. Thanks for your help! I just found another issue with another font I can't figure out. I will post that one shortly. I love the awkwardness of all caps. It's uncomfortable but in an edgy, good way.

Now here is the other typeface. It seems they changed the main headline type with each issue...Does anyone recognize it?

Close to Adobe Caslon, but not exactly. Caslonesque.

Maybe Garamond medium italic?
http://www.myfonts.com/fonts/efscangraphic/garamond-no-2-sb/medium-ita/

VERY close. But the “p” and the “J” are different.

Love that "ry"-ligature. :^)

Sushi, I think that's DTL Elzevir bold italic. See p. 49 of their PDF here. I think I've seen that wild ry ligature in the PDF, but I don't see it right now.

It is related to Caslon as many say that the types of Van Dijck, which are the basis of DTL Elzevir, served as models for William Caslon.

By the way, heinous use of Caslon 540 :)