Wake up MCU on UART RX data
Any MPU with pin change interrupts should allow this. Some (MSP430 at least) can wake in a microsecond and start timing the start pulse  right away. But even if your micro can't wake up fast enough, can you transmit an extra character at the start of the message to wake it up? 

If you give us a clue which MCU you mean, someone might have a more appropriate suggestion.

The datasheet for the LPC11U1x says

(page 2)


  Up to 8 GPIO pins can be selected as edge and level sensitive
  interrupt sources.


(page 14)  


  USART
     RXD ... PIO0_18 ...


so the UART RX pin is also a GPIO pin (PIO0_18 on some packages)

(page 18)


  Any GPIO pin providing a digital function can be programmed to
  generate an interrupt on a level, a rising or falling edge, or both.


(page 26)


  The LPC11U1x can wake up from Deep-sleep mode via reset, selected GPIO
  pins, a watchdog timer interrupt, or an interrupt generating USB port
  activity.


The User Manual says
(page 35)


  3.5.33 Pin interrupt select registers Each of these 8 registers selects one GPIO pin from all GPIO pins on both ports as the source of
  a pin interrupt. To select a pin for any of the eight pin interrupts,
  write the pin number as 0 to 23 for pins PIO0_0 to PIO0_23


(page 38)


  3.5.37 Interrupt wake-up enable register 1 This register selects which interrupts will wake the LPC11Uxx from deep-sleep and power-down
  modes. Interrupts selected by a one in these registers must be enabled
  in the NVIC (Table 58) in order to successfully wake the LPC11Uxx from
  deep-sleep or power-down mode


So it looks feasible. One remaining question is how fast you can get out of deep sleep in the interrupt handler and restart the UART. I'll take a wild guess and say fast enough.