Yeah, Alex, it looks like you need a cast to (LPC_UART_TypeDef) on lines 165 and 166:
UART_Init( (LPC_UART_TypeDef *)LPC_UART1,&uartConfig);
UART_TxCmd( (LPC_UART_TypeDef *)LPC_UART1,ENABLE);
For some reason, in whatever version of the tool you're using, uart1 is treated differently than the other uarts. You can see it here, on line 221 of lpc17xx_uart.c that you included:
if(((LPC_UART1_TypeDef *)UARTx) == LPC_UART1)
All the other instances of uart default to LPC_UART_TypeDef. Make those typecasts, above, and it *should* work. You'll need to typecast LPC_UART1 every time you use it, I'd imagine.