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In quantum chaotic systems, the spectral form factor (SFF), defined as the Fourier transform of the two-level spectral correlation function, is known to follow random matrix theory (RMT), namely a 'ramp' followed by a 'plateau' in sufficiently late times. Recently, a generic early-time deviation from the RMT behavior, which we call the 'bump', was shown to exist in random quantum circuits and spin chains as toy models for many-body quantum chaotic systems. Here we demonstrate the existence of the 'bump-ramp-plateau' behavior in the SFF for a number of paradigmatic and stroboscopically-driven 1D cold atom models: (i) Bose-Hubbard model, (ii) spin$-1/2$ Bose-Hubbard model, and (iii) nonintegrable spin-$1$ condensate with contact or dipolar interactions. We find that the scaling of the many-body Thouless time $t_{\textrm{Th}}$ -- the onset of RMT -- , and the bump amplitude are more sensitive to variations in atom number than the lattice size regardless of the hyperfine structure, the symmetry classes, or the choice of driving protocol. Moreover, $t_{\textrm{Th}}$ scaling and the increase of the bump amplitude in atom number are significantly slower in spinor gases than interacting bosons in 1D optical lattices, demonstrating the role of locality. We obtain universal scaling functions of SFF which suggest power-law behavior for the bump regime in quantum chaotic cold-atom systems, and propose an interference measurement protocol.