Nope. The 4053 is no good for directly switching power.
Personally I'd use a 4052 for the signal lines and P-MOSFETs (with gate-source pullup resistors so they are normally off) as high side switches to control power to the two UARTs. IF you use a 74HCT4052, its inputs are 3.3V compatible when its running with 5V Vcc, so no level conversion required.
If you can rearrange the circuit to have one active high MCU output to enable each UART, rather than having IO2 as a master enable and IO1 selecting the UART it would make it much easier as then you could use
AO6602 complimentary MOSFETs, with its N-MOS device as a level shifter driving the gate of its P-MOS device so the MCU can control a supply voltage greater than its logic '1' level. Alternatively, if there is a higher unregulated rail available, simply use a separate regulator with an enable pin for each UART.
Alternatively, keep the 4053, (use the 74HCT version to avoid the need to use level shifters), and use its third channel to switch ground to the gate of one of the high side P-MOSFETs at a time to switch on that UART. To extend it up to 16 ways, use three
74HCT4067 chips, for the RX, TX and Vcc gate control, and 16 P-MOSFETs and gate pullup resistors.