While waiting for nosr to make the comments here into a full answer, I'll add some references to the literature. The early sources are quite technical and don't provide much in the way of user-friendly examples, but they do show the origins of the ideas involved in Weil restriction:

A. Weil, *Adeles and Algebraic Groups* (Birkhauser, PM 23, 1982): this small monograph contains a verbatim copy of Weil's 1959-60 IAS lectures on Tamagawa numbers, along with some updates by T. Ono and a bibliography. See section 1.3 for the starting formalism of Weil restriction of scalars.

A. Borel and J.-P. Serre, *Theoremes de finitude en cohomologie galoisienne*, Comment. Math. Helv. 39 (1964), 111-164: see section 2.8. [This paper and the following one are available online, the second at numdam.org]

A. Borel and J. Tits, *Groupes reductifs*, Publ. Math. IHES 27 (1965): see especially section 6.

M. Demazure and P. Gabriel, *Groupes algebriques, tome I*, Masson and North-Holland, 1970: see I, section 4, 6.4 and 6.6. (Their framework is more scheme-theoretic than the early ones.)

Much more recently, Weil restriction has played a large role in the book *Pseudo-reductive Groups* (Cambridge, 2010) by B. Conrad, O. Gabber, G. Prasad. In their index, see the many specific topics listed under "Weil restriction". Here I haven't tracked down an explicit comparison of the Weyl groups, but for instance Prop. A.5.15 comes close in its treatment of tori.

My only editorial comment would be that no one over the years seems to have provided an ideal intuitive discussion of what Weil restriction is all about and why it's important. Some explicit examples could be very enlightening, including the three-dimensional simple group $\mathrm{SL}_2$.

ADDED: For a typical application of Weil restriction (in language close to Weil's) which brings out implicitly the connection between the maximal split tori (and Weyl groups) in the groups over the two fields involved, see 3.1.2 in the article by J. Tits, "Classification of algebraic semisimple groups", Proc. Symp. Pure Math. 9, Amer. Math. Soc., 1966 (proceedings of 1965 Boulder summer institute). As in other sources, the emphasis is on comparing tori root systems and rather than directly on the Weyl groups.