How does an ADIRU convert ram air pressure from the Pitot tube into Airspeed?
Usually there is an intermediate module ADM (air data module) that makes the conversion from analog pressure into digital numerical value. One ADM will do this for  the ram air pressure, another ADM will do the same for  the static pressure. 
Their data are sent on Arinc numerical buses to the Air Data part of the ADIRU. 
Airspeed is basically the difference of the received data.
Please refer to the following diagram:

https://upload.wikimedia.org/wikipedia/commons/f/ff/Airspeed_indication_system_-_fly_by_wire.png

Inside the ADM you have:
first of all a PIEZO that will convert the mechanical pressure into electrical voltage, a piezo is somehow what you have in a balance you may use to get your weight. In the balance you have too an analog digital converter since the weight is displayed as a number.

Similarly in the ADM, the next item is an analog/digital converter that will convert the electrical voltage into a binary number that will be sent on the ARINC bus to the ADIRU

You might wonder why the ADM is apart and not inside the ADIRU, it is done like that to have the shortest possible pneumatic tube.
 I do understand I haven’t been enough clear in my previous answer that has been deleted, therefore it is important to understand the following:

The ADIRU managing Air Data and Inertial data, is located in an electronic bay, the pneumatic piping is not sent to the electronic bay, that will make a long distance, therefore no pressure conversion is done by the ADIRU, but by intermediate modules, the ADMs