With the boiler off remove the large slot head screw from the front of the pump, insert a screwdriver into the pump and see if it turns freely, it may be seized. You will get some water dribbling out so have a towel ready.
What's the boiler doing while hot water is running?
Is the flame symbol going out?
Sounds like a circulation issue, blocked secondary heat exchanger perhaps.
Depends on how competent you are at this sort of thing. Looks straightforward, mains water off, boiler off and isolated, drain the boiler, remove and replace.
Plenty of you tube videos on replacing one of these.
I use the Stuart Turner for the shower but to increase hot water flow throughout the house I installed a flow flex sp2b in the 22mm hot water pipe from the hot water cylinder, works a treat.
You can tie a plastic bag over the outlet and see if it fills up. Another thing that can cause pressure variations is the expansion vessel needing recharging with air. There's a how to section on the method in the FAQs
In your second picture you need to unscrew the metal connection from the waste, usually a pair of grips will suffice, try not to bugger up the threads, if there's no room then Disconnect the taps and plastic waste and take the sink off the wall.