The Simons Observatory (SO) includes four telescopes that will measure the temperature and polarization of the cosmic microwave background using over 60,000 highly sensitive transition-edge bolometers (TES). These multichroic TES bolometers are read out by a microwave RF SQUID multiplexing system with a multiplexing factor of 910. Given that both TESes and SQUIDs are susceptible to magnetic field pickup and that it is hard to predict how they will respond to such fields, it is important to characterize the magnetic response of these systems empirically. This information can then be used to limit spurious signals by informing magnetic shielding designs for the detectors and readout. This paper focuses on measurements of magnetic pickup with different magnetic shielding configurations for the SO universal multiplexing module (UMM), which contains the SQUIDs, associated resonators, and TES bias circuit. The magnetic pickup of a prototype UMM was tested under three shielding configurations: no shielding (copper packaging), aluminum packaging for the UMM, and a tin/lead-plated shield surrounding the entire dilution refrigerator 100 mK cold stage. The measurements show that the aluminum packaging outperforms the copper packaging by a shielding factor of 8-10, and adding the tin/lead-plated 1K shield further increases the relative shielding factor in the aluminum configuration by 1-2 orders of magnitude.