Is it possible to connect a peripheral mouse to the USB side of a USB/UART bridge that is controlled by a CP2103 chip already on an FPGA board?
Almost all ICs marketed as "USB/UART bridge" or similar (including the CP2103) can only act as a USB device on their USB port, so no - this won't work.

What you are looking for is a "USB host controller". Those will have a USB host port to interface with your mouse (or another USB device), and a host controller interface to connect to a host processor - typically a CPU of some kind, but in your case an FPGA. In most cases this host controller interface will be an implementation of one of several standards such as UHCI, EHCI, XHCI.

Unfortunately for you, those are very complicated standards and it would take a lot of work to implement the corresponding logic in the FPGA. Fortunately for you, there are a small number of host controllers designed for this kind of application which integrate a lot of that logic and present a simpler, higher-level interface to the host processor. Examples include the MAX3421E and FTDI's Vinculum (VNC1L, VNC2).