I'm not sure how would one go about wiping and reinstalling Ubuntu remotely, but you could try the following to upgrade remotely, without apt and do-release-upgrade.
Download the "Alternate CD" for 12.04.3 LTS server edition using:
For 64-bit:
wget http://releases.ubuntu.com/precise/ubuntu-12.04.3-alternate-amd64.iso
For 32-bit:
wget http://releases.ubuntu.com/precise/ubuntu-12.04.3-alternate-i386.iso
Mount it to /media/cdrom using:
sudo mount -o loop ubuntu-12.04.3-alternate-amd64.iso /media/cdrom
Run the following command to start the upgrade:
sh /media/cdrom/cdromupgrade
Or, if you're not root:
gksu "sh '/media/cdrom/cdromupgrade'"
Note 1: I don't know if it's possible to directly upgrade from 10.10 to 12.04. I think you'll need to go through the releases in between, one by one.
Note 2: I haven't tried the above steps before, so I'm not sure if it will work. But according to this: PreciseUpgrade, it should. The steps there indicate to mount the .iso file to "the exact name of the CD", but I'm not sure what the exact name of the CD is on 10.10.