Point 1: The Church forces in “A Good Man Goes To War” use an Omega symbol as part of their unit insignia.
Point 2: ”The Doctors Wife” revealed that Time Lords can regenerate from male to female and vice versa.
Point 3: ”Human Nature” introduced the idea that Time Lords can hide as human. This happened again in “Utopia” and was referenced again in “The Next Doctor”, suggesting that this was not at all unheard of. Likely uncommonly done, but it doesn’t seem to be particularly novel for a Time Lord.
Point 4: The Master has shown that Time Lords can both inhabit the bodies of other species, and part of his storyline since the 80s was the idea of gaining new regenerations as an actual Time Lord. This was confirmed to actually be possible in “The Sound of Drums”. Apparently, another Time Lord could be killed to provide a new set of regenerations.
Point 5: Omega has tried, more than once, to escape the antimatter universe he was trapped in, and every time, it was The Doctor that stopped him. He has it in for our favorite Time Lord. He’s also tried taking The Doctor’s own identity, copying his body.
So, my theory. Madame Kovarian is Omega. He is either inhabiting a human or near-human host, or is on or close to his last regeneration. Melody/River has two jobs in Omegas plan. First, is to eliminate Omegas greatest enemy, The Doctor. The second, is as a new host body that allows Omega to be a full Time Lord once again. Alternately, a source of energy to grant himself new regenerations.
Why hasn’t The Doctor caught on, having met Kovarian? Well, the one time he’s known to have met her, time was collapsing. This could have blunted his senses. Also, if Omega was inhabiting a cross species host, or had been fobbed, he might not be detectable by a fellow Time Lord. Also, Omega is from Gallifrey’s ancient history, the very dawn of the Time Lord era. Billions of years before The Doctor. The psychic “resonance” or whatever might not have been as strong as it would be for a more modern Time Lord, or might be of a substantially different character. Also, Omegas time in the antimatter universe might have changed him enough to be undetectable.