Even a cursory reading of the spec indicates that this will not work.
Must: Defines the minimum viable product clinically acceptable by clinicians [...]
- RMCPAP must deliver inspired oxygen concentration in the range 35 – 80% to the patient, selectable by the user. [...]
- All gas connectors and hoses must comply with BS EN ISO 5359:2014+A1:2017, ISO 5359:2014/AMD 1:2017 and BS 2050: 1978
Electrical Conductivity [...]
- Must connect to wall pipeline oxygen supply via BS 5682:2015 compatible probes [...]
- Must maintain a nearly constant airway pressure of between 5–15 cmH2O, with the ability to adjust the pressure [...]
- Must either alarm or be provided with a suitable air entrainment system if the fresh gas supply fails [...]
- Must have as low a resistance to expiration as possible [...]
- Expiratory gas must be able to pass through an appropriate filter [...]
- Must have a pressure safety release valve to protect the patient from high pressures, having a release pressure of no greater than 25
- Must be securely attached to the gas outlet
A bike pump does not meet any of those criteria, this isn't even the complete list of "musts", and failing on a single criteria would already mean that the pump is not suitable. Adding enough peripherals to meet even some of the criteria would in all likelihood impossible, and if possible the resulting device would no longer resemble a bike pump.
You could probably use a bike pump to pump air into a lung, but given that a bike pump is designed to deliver up to 160 psi ("Portable pump psi ratings can vary. Some offer 90 psi, others up to 160 psi"), that's just an interesting way to kill a person, which is not usually what ventilators are for.
The closest to this plan I could find was a plan to convert an ambu bag into a ventilator. My experience with ventilators is rather more practical than theoretical, and given what fickle machines even state of the art ventilators are, I would probably choose death over that sort of improvised machine.