Sink blocking

WIAWSNB

Hi SarcyUsername.
Just how DIYish do you reckon you are? 
This is a nice starter project to get you used to 'waste' plumbing, as everything is hand-tight, and needs no special tools. Also, it should all go back the same way as it's taken apart, so no mods required.
Having said that, you may need waterpump pliers if some collars are tight, and a tub of silicone grease would be useful.
Are you up for removing all that visible waste to check it all out? It's currently a mystery as to the cause of the problem, and equally so that the W/M appears to shift it. Yes, the WM water is pumped out with greater force than the mere gravity pressure from the sink content, but still. 
The topmost waste connector to the large part that holds the plug looks as tho' its thread could be 'crossed', although it isn't clear. But the 'overflow' pipe - black, and coming down the side - is definitely leaking.
So it all looks as tho' it would benefit from complete dismantling, cleaning, new washers if needed, silicone lube, and reassembly - whilst also hopefully finding the cause of the blockage.

twopenny

I still keep an actual plunger.
Great fun.
Put 2 or 3 inches of water in the sink, pull plug out and quickly press plunger over. Push down hard and pull up several times.
Take all your aggression out on it (doubles as therapy) and pull out sharply with a huge plop.
You may need 2 or 3 goes to really do a thorough job.
That should release anything 

What I did wonder is if that's a wire for a pop up plug that isn't there any more, the wire may have broken off and be hanging around twisted in the pipe.