We study nuclear embeddings for spaces of Besov and Triebel-Lizorkin type defined on quasi-bounded domains Ω⊂Rd. The counterpart for such function spaces defined on bounded domains has been considered for a long time and the complete answer was obtained only recently. Compact embeddings for function spaces defined on quasi-bounded domains have already been studied in detail, also concerning their entropy and s-numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has already been introduced by Grothendieck in 1955. Our second main contribution is the generalisation of the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of ℓr type, 1≤r≤∞. We can now extend this to the setting of general vector-valued sequence spaces of type ℓq(βjℓpMj) with 1≤p,q≤∞, Mj∈N0 and a weight sequence with βj>0. In particular, we prove a criterion for the embedding idβ:ℓq1(βjℓp1Mj)↪ℓq2(ℓp2Mj) to be nuclear.