I guess you are referring to the Tits Bourbaki seminar talk #119 *here* in the 1950s on subalgebras of semisimple Lie algebras (which translates the original question about compact Lie groups)? That's freely available online, but Dynkin's earlier papers probably aren't (though AMS published them long ago in translation volumes). Apparently Dynkin's tables contained some errors, as tables often do, but there is a useful AMS/IP collection *here* with corrections and commentaries which may be the best current source for the full story.

I suspect there has been too little incentive for anyone to rework and publish all of Dynkin's arguments and tables, but Gary Seitz (along with Martin Liebeck) did publish a number of long papers generalizing this work to the setting of semisimple algebraic groups (giving a lot of details in a modern style).

ADDED: There are two important (and long) papers by Dynkin in 1952, published by AMS in Series 2, Volume 6 of their translation series (1957), pages 111-378: *Semisimple subalgebras of semisimple Lie algebras* and *Maximal subgroups of the classical groups*. The papers contain lots of tables and are related to each other, as discussed by Tits in his Bourbaki talk. The main results are recovered by Seitz (and Liebeck) in their large AMS Memoir papers (such as No. 365 in 1987) treating maximal subgroups of semisimple algebraic groups. But this is a more elaborate framework, getting into prime characteristic as well.

To apply Dynkin's results to compact Lie groups one has to use Weyl's approach: complexify the Lie algebra or in reverse take a compact real form of a complex Lie algebra. Since a compact connected group is semisimple (or trivial) modulo its center and the latter lies in a torus, the semisimple case is crucial for maximal subgroup problems. The relevant maximal subalgebras of an associated complex Lie algebra are then semisimple. Unfortunately, compact groups don't seem to be treated explicitly in the literature (an exposition with some low rank groups as examples would be useful). But studying the group structure directly is too difficult, so the Lie algebra technology over $\mathbb{C}$ is most natural here, including some finite dimensional representation theory.