You are wise to ask because this is an injury that you want to heal well. 3 weeks is not very long into the healing of a fracture.
Most references that I looked up tell you that your therapist will direct your safe return to activity. A PT (physical therapist) or OT (occupational therapist) who specialize in hands (hand therapist) are the best ones to advise you. I would ask them for advice about pressure on your hand, wrist and thumb when holding on to the handlebars or braking. I would also ask what type of splint they recommend that can give you protection during activities until fully healed. It is well worth a visit. In most states you can go directly to the therapist and don't need your doctor to refer you. Do take your xrays.
The Sports Injury Bulletin states:
The end point of treatment must be the same as for other fractures, ie, radiographic and clinical union, to allow full function and prevent post-traumatic arthrosis. In athletes, however, it is necessary to ensure a rapid return to participation in sports. For an undisplaced scaphoid fracture the average union rate is approximately 90% with a healing time of 9-12 weeks
Maybe the safest thing to do at this point is spinning. You won't have to worry about falling off and re-injuring and you can maintain your fitness for cycling.
The AAOS, American Association of Orthopedic Surgeon's site, OrthoInfo points out that fractures of the scaphoid at different locations require different treatments and recover differently. They recommend that during healing you should:
•Avoid heavy lifting, carrying, pushing, pulling, or throwing with the injured arm
•Do not participate in contact sports
•Do not climb ladders or trees
•Avoid activities with a risk of falling onto hand (for example, inline skating, jumping on a trampoline)
Best of luck.