Why am I not getting full water pressure, it is still running a third of water through the tub faucet...could it be the filter screen

Although you have not shared the make, model, and year, I will give it a try anyway.
Start by turning off the water at the faucets for the washer.
Then unhook the hose ( or both of them) and attach a garden hose to the faucet, and run the hose outside, or if that is not possible, run the hose to a sink or a shower or a tub, for the drain.
Turn on the water faucet and see if you get good pressure through the water hose.
If you do, then examine the black hoses from the faucets to the washer.
Often these have a tiny wire mesh filters inside the end or ends of them to filter the incoming water.
These can plug up with sediment, and you will need to clean them.
If they are clear, then examine the attachment on the back of the washer, where the hoses go.
These can also have tiny wire mesh filters inside, that can plug up with sediment.
Those screens can be removed and cleaned.
God bless your efforts.

There's a little filter in faucet head and it may have got backed up with calcium from old water heater when changed.. U can just pop it out with lil knife and then run faucet with hot water to get any build up.. U can pick up more screen filters at local hardware store for cheap..

You probably have a water flow reducer in the shower head. It is a rubber disk with a few holes in it. Just remove it and put the shower head back on. They are installed at the factory to save our environment. (Mine are all removed)

There are two possible problems:

1) The crud you don't want in aerator can get stuck in elbows and shut-off valves.
Open following link for steps
http://waterheatertimer.org/Low-hot-water-pressure.html

2) Crossover. If single-handled faucet has bad cartridge, or check valve is bad, these things can cause hot water not to flow.
Generally however if the pressure is still good, the issue is most likely #1 above.
http://waterheatertimer.org/pdf/Crossover-on-single-handled-faucet.pdf

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It is very common when you first install any faucet, for the aerator on the very end of the faucet (where the water comes out into the sink) to plug up with rust or sediment. Because you had to turn off water supply, it churns up this sediment in pipes. It initially plugs up the filter/screen in aerator. Just unscrew the end & flush it out with water. Then reattach.

Check to see if there are any screen type filter to the tub as debris may be clogging it. Also air lock can occur but it should not be causing problems as 90% clear them selves.....

no the tub faucet is the only faucet that is wide open you might have to not run full blast on the faucet , your tankless water heater is under sized for the application, to many gallons per min going thru the unit, look at the rating chart for your unit, the r94 is only 9.4 gallons per min, and a tub faucet can be upto 5 gallons per min, depending on the faucet,,

it's the thermostatic valvle to be regulated

When you installed the new tub faucet you probably turned off the water to that faucet of even the whole house. IF IT IS AN OLDER HOUSE, TURNING THE WATER OFF AND BACK ON CAN LOOSEN SOME DEBRIS IN THE WATER LINES THAT CAN FIND THEIR WAY TO YOUR NEW FAUCET. If there is a screen at the end of the water outlet it can be clogged with gunk. Uncrew the screen on the water outlet and clean. You may also want to flush out the hot and cold outlets by turning the water off, removing the cartigages and slowly let water flush them out. Then reassemble. Also when you have the tub handles out you can turn the hot and cold supply valves off and on a few times to loosen more debris if it exsists........Joe

If the washer doesn't fill properly, or if only one temperature of water fills the tub, something is probably keeping water from entering the machine or it's draining before the tub fills completely. Set the controls to the first wash cycle, and then try each water temperature setting. If both hot and cold water seem to be entering the washer with good pressure, water may be draining out prematurely during the fill cycle, through siphoning action. Be sure the drain hose's connection to the standpipe is at least 34 inches above the floor (otherwise, discuss the problem with a plumber). An air gap between drain hose and stand pipe that prevents backflow is also good to have. If hot works but cold doesn't, or vice versa: 1) Be sure both the hot and cold water faucets are turned on. 2) Be sure the supply hoses are not kinked. 3) Turn off the faucets. Unscrew the hoses from the faucets and drain them into a bucket. Hold the bucket under the faucets and test them to make sure they both work and have plenty of pressure (see HERE for information on replacing faucets). Screw the hoses back onto the faucets. 4) Unscrew the hoses from the inlet valves and clean the filter screens just inside them; these may be clogged. To do this, pry out the screens with a screwdriver (this can be difficult; the screens may be inside the valves), flush them under running water, then put them back into the hose. If the screens look defective, replace them. Then screw the hoses back onto the valves. Be careful not to cross-thread the hose threads when screwing them back onto the valves. 5) Call an appliance repairperson; there may be a defective inlet valve or a problem with the controls.