I refer here to my recent answer to
What is the intuition behind the Freudenthal suspension theorem?
where the results do not require simple connectivity for descriptions of the critical group, basically because the proofs do not use homological, i.e. abelian, methods.
May 30: The original Blakers-Massey results were related to triad homotopy groups, since the exact sequences involving these and relative homotopy groups showed the triad groups as the obstruction to excision. So there was a question of calculating these groups, and homology groups were used for this in the simply connected case, see the book by J.F. Adams A student's guide to algebraic topology. However such calculations in the non simply connected case do follow from a Generalised Van Kampen Theorem proved with J.-L. Loday. I have revised and updated a paper of mine ``Triadic Van Kampen theorems and Hurewicz theorems'', Algebraic Topology, Proc. Int. Conf. Evanston March 1988, Edited M.Mahowald and S.Priddy, Cont. Math. 96 (1989) 39-57.
and made it available as
http://pages.bangor.ac.uk/~mas010/pdffiles/VKTEVAN2.pdf
June 9, 2017 There is a nice book on "Cubical Homotopy" by Munson and Volic (CUP, 2015) which deals with a lot of these connectivity arguments. I am unclear whether these argument cover the connectivity results of Theorem 6.1 of
Brown, R. and Loday, J.-L. ‘Homotopical excision, and Hurewicz theorems for n-cubes of spaces’. Proc. London Math. Soc. (3) 54 (1) (1987) 176–192,
which gives algebraic as well as connectivity results.