Python 2, 40 bytes

f=lambda a:a>[]and max(len(a),*map(f,a))

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Uses Python 2's global ordering: lists are all considered greater than ints. We can also use > instead of >=, because []>[] is False which is the same as 0, which would be the result anyway from the max(len(a),*map(f,a)). Also uses the var-args version of max which is shorter than constructing a list.

Whython, 34 bytes

f=lambda a:max(len(a),*map(f,a))?0

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When a member is an integer, the len call fails; we just catch this and replace the result with 0.

Python 2, 40 bytes

f=lambda a:a>[]and max(len(a),*map(f,a))

Attempt This Online!

Uses Python 2's global ordering: lists are all considered greater than ints. We can also use > instead of >=, because []>[] is False which is the same as 0, which would be the result anyway from the max(len(a),*map(f,a)).

This also uses the var-args version of max which is shorter than constructing a list. Normally we wouldn't be able to do this, because the var-args form of max only works with 2 or more arguments, but if we know a>[] then a is non-empty, so *map(f,a) always adds at least one more argument to the call.

Whython, 34 bytes

f=lambda a:max(len(a),*map(f,a))?0

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When a member is an integer, the len call fails; we just catch this and replace the result with 0. When a is an empty list, this also fails because max gets only one argument, but catching and returning 0 conveniently fixes this too.

Python 2, 40 bytes

f=lambda a:a>[]and max(len(a),*map(f,a))

Attempt This Online!

Uses Python 2's global ordering: lists are all considered greater than ints. We can also use > instead of >=, because []>[] is False which is the same as 0, which would be the result anyway from the max(len(a),*map(f,a)). Also uses the var-args version of max which is shorter than constructing a list.

Python 2, 40 bytes

f=lambda a:a>[]and max(len(a),*map(f,a))

Attempt This Online!

Uses Python 2's global ordering: lists are all considered greater than ints. We can also use > instead of >=, because []>[] is False which is the same as 0, which would be the result anyway from the max(len(a),*map(f,a)). Also uses the var-args version of max which is shorter than constructing a list.

Whython, 34 bytes

f=lambda a:max(len(a),*map(f,a))?0

Attempt This Online!

When a member is an integer, the len call fails; we just catch this and replace the result with 0.