Based on the multi-component phase field theory, in this paper we propose an axisymmetric lattice Boltzmann model for three-phase fluids. The proposed model takes advantage of two particle distribution functions for capturing phase interface among three different fluids, and another particle distribution function for solving the hydrodynamic equations for flow field. In order to describe the axisymmetric effect arising from the coordinate transformation, we elaborately design the equilibrium distribution function and forcing distribution function in the evolution equation, which ensures that the model can accurately recover the macroscopic governing equation for three-phase fluids. Also, the introduced source terms accounting for the axisymmetric effect contain no additional gradient term, which makes it be simpler than the existing lattice Boltzmann model for axisymmetric three-phase fluids. To validate the proposed model, a series of axisymmetric multiphase benchmark examples are performed, including the static double droplets, the spreading of liquid lens, and the binary-fluid Rayleigh-Plateau instability. It is reported that the present model can accurately capture the phase interface, and the predicted steady shapes of the liquid lens agree well with the analytical profiles. Then, the proposed model is used to study the three-phase Rayleigh-Plateau instability and the effects of the wavenumber and the radius ratio of liquid column on the interfacial dynamic behaviour, the breakup time of liquid threads and the size of daughter droplet are investigated in detail. It can be found that the compound liquid thread at a high wavenumber could break up into one main droplet and one satellite droplet, but the multiple satellite droplets can be produced at a low wavenumber, which leads to that the sizes of main and satellite droplets increase with the wavenumber at first and then decrease with it. Besides, we can observe that the inner fluid undergoes the breakup at earlier time than the middle fluid, and the breakup time for both inner and middle fluids increases with the decrease of the wavenumber. Finally, we can find that increasing the radius ratio of liquid column accelerates the breakup of inner-fluid thread, but prevents the breakup of the middle-fluid thread. In addition, the size of the compound main droplet increases with the radius ratio of liquid column, while the size of the compound satellite droplet doest not change much with it.