Dilute polymer solutions are known to exhibit purely elastic instabilities even when the fluid inertia is negligible. Here we report the quantitative evidence of two consecutive oscillatory elastic instabilities in an elongation flow of a dilute polymer solution as realized in a T-junction geometry with a long recirculating cavity. The main result reported here is the observation and characterization of the first transition as a forward Hopf bifurcation resulted in a uniformly oscillating state due to breaking of time translational invariance. This unexpected finding is in contrast with previous experiments and numerical simulations performed in similar ranges of the Wi and Re numbers, where the forward fork-bifurcation into a steady asymmetric flow due to the broken spatial inversion symmetry was reported. We discuss the plausible discrepancy between our findings and previous studies that could be attributed to the long recirculating cavity, where the length of the recirculating cavity plays a crucial role in the breaking of time translational invariance instead of the spatial inversion. The second transition is manifested via time aperiodic transverse fluctuations of the interface between the dyed and undyed fluid streams at the channel junction and advected downstream by the mean flow. Both instabilities are characterized by fluid discharge-rate and simultaneous imaging of the interface between the dyed and undyed fluid streams in the outflow channel.