I refer here to my recent answer to 


https://mathoverflow.net/questions/56435/what-is-the-intuition-behind-the-freudenthal-suspension-theorem/85158#85158

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.