During cold season almost all the time we need to wait for hot water coming from the water heater to our faucets, showers etc. I want to install tankless water heater in the cold line as a supplemental element. In other words it is going to warm up the cold water in the cold line to reduce the wait time for getting hot water from the conventional water heater. Of course during hot season we can turn off the tankless water heater. In other words this tankless water heater will only be used on demand with lesser capacity because conventional water heater's hot water will also be mixed in the end points such as shower, faucets etc. Will this work?
I really don't understand your proposal but if you're adamant about it then just supplement your hot line with a tankless closer to your fixtures. If you set your tank to 125F and your tankless to 120F then your tankless should only operate for a very short period instead of always warming the incoming cold line. If your tank ever runs out then your tankless will pick up the slack.
Better yet, relocate your water heater closer to your fixtures; this would save you thousands as average cost for tankless install is $4,000.
I see your proposal has a third line going from the tankless to your fixtures but fixtures don't accept a third line. You would have to junction it into one of the lines (presumably the hot) right before the fixture. Even if you junction into a hot line it will be a mix of hot (tankless) plus cold (tank) at the hot tap until both lines are full of hot water so this would actually increase the total wait time for hot water at the tap.
One thing you cannot overlook is that tankless units are typically 200,000 BTU/HR and typical residential natural gas can supply about 225,000 BTU/HR so a gas upgrade will probably be required.
"Ditto", to what the others have said. In order to get hot water at the "point of use", the faucets have to expel the cold water that is in that line. Adding a tankless heater would be a very expensive way to solve that problem. Adding it to the cold water line would not allow you to control the temperature of water coming out of the spout. You would have only hot or hotter water with no way to adjust the temperature for the person using that water. Point of use heaters would solve that problem but adding one to each faucet would also be expensive.
I did what @George Anderson suggested and added a recirculation line from each of my 4 bathrooms when my house was built 22 years ago. I control it's operation with a cheap timer. To make mine code compliant, since I am not an electrician, I added a plug to the electric cord and plug the timer into a wall socket and plug the pump into the timer. No registered electrician needed. I had a 1/2" tee installed as close to the bathrooms as possible, added an isolation valve and check valve to each line. All the lines came back to a manifold at the water heater, were run through the pump, and the outlet of the pump runs into where the tank drain was. The pump must be a bronze or stainless steel to resist the corrosion of the fresh water on a steel pump. Buy the smallest pump you can get. Mine is a B&G bronze rated at 150PSI and uses 39 watts of electricity . Hope this helps!