I want to sample all 6 channels on the LPC1768 at the same time, at 5kHz.
How does this work? According to the datasheet it has 1 ADC with a multiplexer.
So if I make all 6 pins an AnalogIn, it will not be possible to sample all of these at the same time?
The best I could hope for is perhaps within 200us or so or one another because I’m reading that it takes about 22us for the Analog read function to perform, then storing the data for each one etc
Am I right about this?
Yes you’re right that each analogue input must be sequentially sampled via the mux.
If you really* need to sample 6 inputs at exactly* the same time then you’ll need an external circuit to implement 6 channels of sample-and-hold. Then read & store the S/H outputs.
Your concern about whether all 6 channels can be scanned in under 200mS is probably real 
Another option is 6 external A/D converters - the ones with SPI or similar interfaces would minimise the number of data lines required. I’d say S/H would still be required if your sampling requirement is set in stone.
*review & verify the requirements in case there’s a workaround