Yes, you could change the advertising payload of the BLE NUS example, sure. You will need to keep advertising the NUS UUID for the example to function as intended - since the central will look for an connect to devices that is advertising the NUS UUID - but you could drop the Nordic_UART device name, and replace it with a manufacturer specific datafield, which contains the company identifier.
You will need to change this in the advertising_init function. You could take a look at this blogpost which includes an example that demonstrates how you may include the manufacturer specific datafield in your advertising payload. The example is made to show how to dynamically update the payload, which is not important in your case, but it still adds the manufacturer specific datafield to the advertising. Please keep the hard 31 byte limit (including 2 bytes overhead per datafield) to the advertising payload length in mind.
