There are two aspects of your question.
Hence you have to be specific that does fast mean, you want to transfer a given amount of thermal energy by least amount of time or you just want to say which mode gives the maximum speed of disturbance or information.
Honestly Speaking the 2nd aspect is not of much concern to engineers. But surely, the answer to the 2nd aspect is radiation. Simple reason is radiation is electromagnetic wave travelling at the highest possible speed. Example is Sun ray coming through space and falling on earth. It takes 8.5 minutes approx to cover the distance and governed by radiation. But it does not suggest that at a given time the maximum heat is transferred from sun to earth. The energy (not the maximum) travells faster by electromagnetic waves. Hence, 2nd aspect is not a matter of thought, engineers mean the fastest way to transfer heat is optimising the thermal resistance so that maximum heat can be transferred in a given time which means higher rate of heat transfer.
Honestly speaking it depends on various situations. Normally, forced convection is considered one of the fastest way to transfer heat. Here we use some blower or pump and flow of fluid stream (advection) is responsible for maximum heat transfer. Example is if you touch something hot immediately you blow air over your hand. Since air both conducts and advects the heat. Another example is fins in cylinder head where maximum surface area of fin enhances more heat transfer by convection. Another example is cooling of Journal bearing, where the rotation of rotor causes the lubricating film to form and transfer the heat by convection. Another examples is flow of fluid in any heat exchangers.
But there are also cases where installing pump or blower might prove to be difficult (Like cooling of chips of I phone), here natural convection is good enough to serve your purpose. And yeah sometime fast transfer of heat can be done by radiation even for small temperature differences (70–80 deg C say). For a vertical plate at 100 deg C it can be seen that radiation contributes to 55% of rate of heat transfer. For high temperature system, radiation transfer heat fastest, Example Superheater section of a boiler (maximum heat is transferred by radiation). There are many parameters by which the relative dominance of one mode of heat transfer over other can be understood. Like Nusselt number is a comparison between conduction and convection. Grasoff and Reynold’s number can be compared to understand between natural and forced convection. Conduction/Convection-radiation parameter suggests the same thing. Hence a direct answer can not be provided, we have to calculate and see which mode is transferring thermal energy faster.
Always remember that in heat transfer we never calculate net heat transfer exclusively. We are generally interested in the rate of heat transfer (i.e. how fast heat can be transferred) and which conditions influence that such that net thermal energy can be transferred at fastest rate.