We study the consequences of having translational invariance in space and in time in many-body quantum chaotic systems. We consider an ensemble of random quantum circuits, composed of single-site random unitaries and nearest neighbour couplings, as a minimal model of translational invariant many-body quantum chaotic systems. We evaluate the spectral form factor (SFF) as a sum over many-body Feynman diagrams, which simplifies in the limit of large local Hilbert space dimension $q$. At sufficiently large $t$, diagrams corresponding to rigid translations dominate, reproducing the random matrix theory (RMT) prediction. At finite $t$, we show that translational invariance introduces an additional mechanism which delays the emergence of RMT. Specifically, we identify two universality classes characterising the approach to RMT: in $d=1$, corrections to RMT are generated by different translations applied to extended domains, known as the crossed diagrams; in $d>1$, corrections are the consequence of deranged defects diagrams, whose defects are dilute and localized due to confinement. We introduce a scaling limit of SFF where these universality classes reduce to simple scaling functions. Lastly, we demonstrate universality of the scaling forms with numerical simulations of two circuit models and discuss the validity of the large $q$ limit in the different cases.
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