It is strange but after you flush the toilet the tank does not fill back up on its own. If I shut of the water valve to the toilet then turn it back on it begins to fill up again. Any ideas on what might be causiing this?

Which type of fill valve do you have?

I have the same question, and have the first image tank. The loo is built-in too, so I've had to remove the panel and have limited ability to see what's inside (covered by a bathroom countertop).

I just jiggle all sorts of things and it fills. I just don't know what I'm touching that makes it work. And really need to get the cover back on.

Thank you in advance

From your description I am guessing that you have a sink counter top that extends over the toilet. Something we in the US would call a banjo counter top. That counter top limits your view of the inside of the toilet tank (cistern).
I am also guessing that after you flush the water keeps running but the tank does not refill.

I just jiggle all sorts of things and it fills.
My next guess is that you are jiggling the toilet handle since there is nothing else to jiggle.

If all my guess are correct, you have a flush valve (flapper) that is not falling down to seal the tank after you flush. If there is enough room between the counter top and the top of the tank to get your arm in, and you know what you are doing, the flapper valve can be replaced. If you can't get your hand into the tank, the toilet will have to be taken up to repair.

A better description of what is and is not happening would helpful. Try using a mirror to observe what is happening.

Thank you for this. I'm just touching things and lifting whatever I can up and down, and wiggling the handle, the inside bits, lifting the 'tube' up and down, whatever is moveable.

There is no handle per se, but a push in dual (low/high) button. That's connected to a rubber wire, which seems connected to the flap you mention. I tried to tighten the button (there's a sort of washer thing on the back of the panel, which is now off the cabinet - can't go far because of the tension), but it doesn't seem to help.

A silly question I'm sure, but are these flappers a standard size? As in, to replace?

Not at all familiar with the push button type of flush mechanism but logic tell me that when you push the button that causes a lever inside the tank to move up, pulling the chain or wire connected to the flapper valve. Push the button and you should be able to see the action.
When you release the button the lever should fall back down and allow the flapper to close.
It is most likely this mechanism that is defective.

Can you post a picture of the button inside the tank?

Not at all familiar with the push button type of flush mechanism but logic tell me that when you push the button that causes a lever inside the tank to move up, pulling the chain or wire connected to the flapper valve. Push the button and you should be able to see the action.
When you release the button the lever should fall back down and allow the flapper to close.
It is most likely this mechanism that is defective.

I will have it replaced.
Thank you so much.

https://www.google.com/search?q=push+button+toilet+flush&client=firefox-a&hs=f1c&rls=org.mozilla:en-US:official&tbm=isch&tbo=u&source=univ&sa=X&ei=kwhkUsHeBe7G4AOSwYBA&ved=0CFkQsAQ&biw=1024&bih=442
Various types of push button dual flush mechanisms.

https://www.google.com/search?q=push+button+toilet+flush&client=firefox-a&hs=f1c&rls=org.mozilla:en-US:official&tbm=isch&tbo=u&source=univ&sa=X&ei=kwhkUsHeBe7G4AOSwYBA&ved=0CFkQsAQ&biw=1024&bih=442
Various types of push button dual flush mechanisms.

Ah, it's this type exactly (front cabinet, not top of tank though): Ideal Standard Twin Pushrod Dual Flush Button - Plumbers Mate Ltd (http://www.plumbers-mate-sales.co.uk/ideal-standard-twin-pushrod-dual-flush-button---08000668-299-p.asp)

And the interior looks to be Homebase - Toilet Cistern Pushbutton Conversion Kit customer reviews - product reviews - read top consumer ratings (http://reviews.homebase.co.uk/1494-en_gb/308820/reviews.htm)

Should be an easy repair.
Observe levers inside toilet. If they fall after pressing push button then problem will be flapper valve. If levers do not fall problem will be with push button mechanism.

Replacing flapper should be easy. The flapper should just hook on a couple hooks on the side of overflow tube. You should be able to remove the old one and install the new one just by feel even if you can see it.

Replacing the push button should also be easy. Unscrew the nut on the inside and pull the mechanism out through the hole.

After installing new mechanism adjust chain so that there is about one inch of slack in the chain.

Note: In US with the lever type flush the nut on the inside is a reverse thread. That is, it screws on in the direction that most nuts screw off. It is done that way so the the nut does not come loose from use of the handle. This may or may not be the same with a push button flush.