grammar - Can placements of adverbs be altered freely?

I'm curious if there is any difference in nuance between these two sentences:

1. 彼は少なくとも週に一度車を洗う。


2. 彼は週に少なくとも一度車を洗う。

I'm aware that grammatically speaking both are 100% right, but this question is not targeted at this issue.

Answer

Your sentence 1 is ambiguous with respect to the scope of 少なくとも:

彼は[少なくとも週に一度]車を洗う

'He washes his car at least once a week.'

彼は少なくとも[週に一度車を洗う]
'He at least washes his car one a week. (He also changes the motor oil once a month.)'

The first meaning is the same one as your setence 2, but the second meaning cannot be expressed by sentence 2.

In general, the position of an adverb often gives different possibility for its scope.