No, they won't. All Pure radios store settings like alarms and presets indefinitely as the radio writes these details to an EPROM chip which doesn't require any power or batteries to keep them stored.

You will need the radio mains powered if you want your timers or alarms to trigger. Our alarms will not sound if you are running your radio on batteries/ChargePAK. ¹

¹ With the exception of Evoke 3, Move 400D, Evoke D4 and Evoke D6