Hi,
I think it is a bit complicated that you want to use uart0 to boot up, you have to configure the board so that it can boot from uart0.

If you want to boot from uart0, all the P1_1/P1_2/P2_8/P2_9 must be low.
the P1_1-BOOT0_LED-SSP0_MISO must be LOW, you can close the pin 3 to pin5 of the J3 connector to make it low. You have to close the JP3 so that P2_8-BOOT2_TRGT is low.
Regarding the UART0 signal, if you J5 USB connector, the PC will enumerate the J5 USB as a CDC port and give a virtual port, you can use the port as UART0 to download code. If you do not connect the J5 USB connector, you have to connect the UART0 via P4 connector.
After the above configuration, you can hold on the SW2 during the power power off then on, after a while, release SW2, then check if you can use FlashMagic to communicate with the LPC4337.
Hope it can help you
BR
XiangJun Rong