My water heater is about six years old and I have never cleaned sediments from the bottom of the tank. The water heater is on the second floor inside a closet. Just below, we have our master bedroom bathroom which has two separate sinks inside a countertop of a vanity. Across from the vanity, there is a standing shower in a small enclosure. For the last several months, when my wife is taking shower, the hot water pressure is fine. But if I turn on the counter sink hot water faucet while shaving, the hot water pressure inside the shower becomes very low so I have to turn off the sink hot water until she is done with her shower. I also tested the pressure difference at the second sink. The shower hot water pressure reduces when we use hot water at any of the sinks. But, when we use only cold water at the sink faucet, the shower hot water pressure is not affected. Does this have anything to do with the accumulation of sediments at the water heater tank. Please advise what causes this pressure reduction and how to resolve this problem.

Flushing the hot water tank won't fix it, if pressure is good elsewhere in the house. It's just the way your bathroom/house is plumbed

Sediments in the tank won't cause pressure problems and draining it can cause other problems. The valves on hot water tanks are cheaply made, at best, and when you drain the tank sediments can cause the washer in the valve to fail due to the sediments that are flowing past it and building up when draining the heater. A slow leak may easily begin at the valves washer and seat and the first thing people do is try to over torque it to stop the drip and then break the often plastic valve body and cause more issues than the sedimant ever did! It would take many years of sedimant to cause an issue with a heating element in an electric water heater. If the heater is gas fired and has a brass valve on it then draining it to reduce sediment may be a good idea but be ready to clean out the valves washer and seat after doing so. The issue you have with the hot water sounds like the plumber ran all of your bathroom on the same 1/2 inch pipe and the diameter of the pipe won't allow for proper flow when all used at the same time. If done properly the bathroom would have been on 3/4" and each faucet branch off with 1/2" pipe and then being reduced to 1/4 at the stop on your vanity. Does the shower head have the water saving reducer still in it? Many times people remove these thinking that the water pressure will be better without it but the opposite is true.