Is it possible to work out the hash using length extension attack but with different hashes type?
This is not possible in general, or for the specific case of SHA1 and SHA512. Length extension works because MD-hashes iterate the same function, so knowing the output you also know the intermediate state for some related longer inputs.

Now there are some cases with "different" hashes where it could be possible. That its when the hashes are related. Like SHA512 -> SHA512/500 or another such truncated hash that uses the same IV and compression function.

In that case you can use the normal length extension attack and just truncate the final output.