Is Base64(SHA1(GUID)) still unique like the original GUID? Basically what the title is; GUIDs are unique by design. If you run the GUID through SHA1 and then Base64 the hash, will the resulting string have the same guaranteed uniqueness as the GUID, or not?
Base64 provides a 1:1 transform from input to output (and back again if desired). So if you take a set of unique items and base64 encode all of them they will all be unique.

So the question becomes if you run a GUID through SHA1, will the resulting hash have the same uniqueness as the GUID?

The answer is practically - yes; theoretically not quite. Multiple inputs to SHA1 map to the same output, but in reality if your GUIDs are unique then there is negligible probability that there will ever be a collision leading to loss of uniqueness.