Why would you use a radio button, which is a multiple choice control? I think you mean a Simple button.
Write a script for that button to prevent changing your TimeField after that is a one time set.
Just guessing here, but perhaps our OP wants to select a value in the radio button field and with a second field that logs the time stamp to record the date and time that the value was selected.
A TimeStamp field set to enter the Modification Timestamp might server for that. Or a script performed by the OnObjectModify trigger might be used with set field to log the current time stamp in a time stamp field.