We determine Gr\"{o}bner bases for the presentation ideals of multi-Rees algebras and their special fiber rings. Specifically, we focus on modules that are direct sums of ideals generated by either maximal minors of a two-sided ladder matrix or unit interval determinantal ideals. Our analysis reveals that the multi-blowup algebras are Koszul Cohen--Macaulay normal domains, possess rational singularities in characteristic zero, and are F-rational in positive characteristic.