There is more background but not an answer to your specific question in the paper Modelling and Computing Homotopy Types:I (MCHT:I) of which the published version, and other material from the same volume, is available from Elsevier by clicking on this link.
One has to be clear on basic principles. Homotopy groups are defined only for spaces with a base point: the distinction between this and a space is often neglected.
Singular homology groups are defined for any topological space $X$ and do satisfy a Mayer-Vietoris exact sequence whenever $X= U \cup V$ and, say, $U,V$ are open in $X$. Usually this sequence gives information on $H_n(X)$ only up to an extension of abelian groups: thus it will not necessarily tell you if the answer is $Z_2\oplus Z_2$ or $Z_4$.
By contrast, the various generalised Seifert-van Kampen theorems give colimit information, i.e. complete information, under some kind of openness condition, but only for certain invariants of structured spaces, and only under certain connectivity information.
Thus in dimension $1$ we get a pushout for $\pi_1(X,S)$ where $S$ is a set of base points but only if $S$ meets each path component of the intersection of $S$ with $U,V, U \cap V$. If you really want information on some group $\pi_1(X,s)$ for $s \in S$ you may have to do some hard combinatorial group(oid) theory work, but a choice of some such $s$ may destroy any symmetry information in the original situation, and may reflect the attitude: "the answer has to be a group!". See this discussion for more information.
In dimension $2$ we get information on the crossed module $$\Pi(X,A,S):= (\pi_2(X,A,S)\to \pi_1(A,S)) . $$
again if $X=U \cup V$ with $U,V$ open and with some connectivity conditions, this time in dimensions $0$ and $1$. These are a limitation but when it works you get complete information, not obtainable, otherwise, on the in general nonabelian crossed module, as above, which is also a model of homotopy $2$-types.
It is still further work to calculate homotopy groups from the more global information. It is sensible to get a complete result on gluing homotopy types, rather than on individual homotopy invariants.
Both the Mayer-Vietoris sequence and the Seifert-van Kampen theorems are local-to-global results. However the latter contain nonabelian information, which is unusual. To obtain higher dimensional nonabelian versions of the fundamental group was an aim of the topologists of the early 20th century. One question at issue is: for which problems are the two approaches more suited?
$1)$ the approach indicated above, using strict algebraic structures defined for some structured spaces, or
$2)$ the approach using
weak $\infty$-categories of some form, defined for all spaces.
One of the features of homotopy theory is that indentifications in low dimensions usually strongly affect higher dimensional invariants. One method of tackling this is to use algebraic systems which have structure in a range of dimensions $0,1, \ldots, n$; this has been achieved by using strict structures of the form of higher groupoids.
I hope that helps.