You could try one of the partition manager freeware/shareware programs out there.
Self-booting Partition Logic is one I've used successfully for similar situations. It's available separately or I think with UBCD. Both are freeware/donateware of excellent quality and functionality. See majorgeeks.com
or similar for download website.
You should be able to create and/or create-copy the partitions to a new drive. Something to keep in mind is the freeware/donateware versions often require that the target partition(s) size is equal to or larger than the original partition. Shouldn't be a functional issue, just have to make sure the size of the new drive can accomodate that. (Keep in mind that some manufacturers describe their drive capacity differently, so if you're replacing say an 120GB drive, I'd use a 180GB or larger replacement drive just to be sure.)
I also suggest you make a backup of the old original drive's partitions (as well as possible) even though they might be "bad" before
you do any work with it with other programs. That's just in case you slip up and tell the partition manager program to do something to the wrong (i.e. original) disk. Partition manipulation programs can be very unforgiving.
edit: it's possible
the computer manufacturer tied the recovery program to the original hard drive serial number so it couldn't be transfered. Generally not, but some manufacturers have done some pretty bizarre things to boost revenues over the years. Just something to keep in mind when considering costs and sourcing of the replacement drive. For example, ask yourself if you would want to be able to return the new HDD if you can't get this to work?...or would you be able to retask the drive in another computer?...etc