Is unbounded quantum fanout operation experimentally feasible?
There are two way to see your question :


Is unbounded fanout a reasonable approximation for realistic (quantum) circuits ?
Is there a realistic quantum architecture which is effectively equivalent to the quantum circuit with unbounded fan-out ?


Peter Shor essentially suggested the formulation 1. in his comment above, and I’ll let experimentalists answer this question. 

The question 2. seems strange, but I bring it forward because its answer is positive : In a 2009 paper (arXiv:0909.4673, paywalled version), Dan Browne, Elham Kashefi, and Simon Perdrix showed that Measurement Based Quantum Computing (MBQC), is equivalent up to a classical side-processing of logarithmic depth, to the quantum circuit model augmented with unbounded fanout gates. 

They basically recall that the unbounded parity gate is a Clifford gate and can be therefore be expressed as a measurement pattern of constant depth (arXiv,paywalled PRA) and show that a fanout gate is nothing else than a parity gate “sandwiched” between two layers of Hadamard gates, and can therefore be simulated in constant depth.

Therefore, your question


  Is there any experimental evidence to suggest whether such constant-latency quantum fanout unitary operations are feasible?


is equivalent to


  Is there any experimental evidence to suggest whether Measurent Based Quantum Computing is feasible?


I personally think the answer to the latter question is “yes”, but the opinions on the feasibility of a given quantum computing architecture tend to be subjective...