I have fixed a few Omegas with shot electronics. It can be a number of things that stuff it up. Recently I made the stupid mistake of leaving batteries inside my vintage Omega and they leaked but thankfully I managed to fix the toy with minimal fuzz as all I had to do was sand the dried acid out of the connectors.
Sky Lynx may be the easier toy to fix as he just has corrosion/leaked battery damage on his connectors. You sand or file the corrosion down to the metal and it will likely work again. At worse you will need to replace it with another conductive metal.
When it comes to Omega, a number of things may have happened. Omega's conductors are rather long and not actually attached to the motor but 'placed' on top of it. If the battery connector comes of its frame, it will break the connection to the motor and not work. It is also possible that there is corrosion in the inner side of the connector.
There are a fair few things that could be wrong with those toys as they have rather complex electronics so I can only give you guesses based on the information given.