Non-uniformity in SHA-1 would be major news.

Note that in the description below, I refer to any keyed involution as a block cipher. One may make a semantic distinction, but any keyed involution could be used as a block cipher (though, of course, most involutions would contain trivial cryptographic weaknesses).

SHA-1 is based around a 160-bit unbalanced Feistel block cipher. The input in broken into blocks, where the final block contains padding and a final count of the amount of data processed. A copy of the 160-bit state is made, the 160 bit state is encrypted using a block of the input as a key, and the initial copy is added back (without carries between 32-bit words) to the original copy. This is repeated for each input block in turn. This is called a Davie-Meyer construction for making a hash function out of a block cipher.

For any Davies-Meyer hash function, the block cipher is invertible and therefore unbiased. The addition is invertible and unbiased. Any bias would therefore have to come from non-zero correlation between addition and encryption. For any moderately complex block cipher, these correlations would be very complex. Real world design of Davies-Meyer hash functions focuses on absolutely minimizing any patterns present, and cryptanalysis focuses on characterizing and approximating any and all minute patterns that escape the design process.

There are some patterns (weaknesses) in SHA-1, but all known weaknesses are way more complex (and minuscule) than could explain the sort of bias seen in this data set, so the bias must be coming from a higher-level source than SHA-1 itself.

On a side note, the addition in Davies-Meyer is to intentionally make the round function non-invertible. If the round function were invertible, there's a trivial birthday attack on the intermediate state between rounds that square roots the strength of the hash function. MD4, MD5, SHA-224, SHA-256, SHA-384, and SHA-512 are all Davies-Meyer constructions using unbalanced Feistel ciphers. RIPEMD-160 is a parallel application of two Davie-Meyers hashes with different initial values, followed by XORing the two outputs to obtain the final output. SHA-3 is the most notable cryptographic hash function that's not a Davie-Meyer construction.

In case you're wondering, one could make a Davies-Meyer hash function using AES. The designers of AES took AES, doubled the word size, doubled the number of words, and fixed a deficiency discovered in the nonlinear byte substitution. The resulting hash function is called Whirlpool, and the underlying block cipher is called Anubis. I'm not aware of any use of Anubis outside of Whirlpool.

The Salsa/ChaCha families of stream ciphers and the Blake family of hash functions are all very similar to each other. They all use a very similar family of (unnamed) block ciphers internally that are twice the size of the desired output. They achieve non-invetibility by breaking the block cipher output into two halves and XORing the two halves together.

Before MD5 was broken, I did read briefly about an attempt (not by Ron Rivest) to use the inner block cipher from MD5 for encryption, but the performance wasn't competitive. Now we've characterized the hidden patterns in the block cipher well enough to break it relatively easily. I forget the name the authors retroactively gave to Rivest's inner block cipher from MD5.