In the recent years, nanoscale superconducting quantum interference devices (nano-SQUIDs) played a fundamental role in the study of small spin systems. Nano-SQUIDs typically employ nano-Dayem bridges having dimensions (length
L and/or width
W) greater than coherence length $\ensuremath{\xi}$ of the superconducting film. They exhibit characteristics different from those of standard SQUIDs because the current-phase relationship (CPR) is nonsinusoidal, rendering most of the theoretical predictions based on the standard SQUID theory usually unreliable. Here, we present a noise theory of dc nano-SQUIDs based on Dayem nanobridges. We have computed the main characteristics of this quantum device including current-voltage and voltage-magnetic flux characteristics, magnetic flux-to-voltage transfer factor, and spectral densities of voltage and magnetic flux noise for
L/$\ensuremath{\xi}$ ratios ranging from 1 (sinusoidal limit) to 3.5 (hysteretic limit). The CPRs have been computed by using the theory of Josephson weak links based on the Ginzburg--Landau equation. The results show a dependence of the magnetic flux noise spectral density on (
L/$\ensuremath{\xi}$)
4/3 involving a degradation of about a factor of five between the two extreme cases and are consistent with experimentally measured magnetic flux noises reported in the literature. These results provide useful information for both device physics and their applications.