Advice desperately needed: no hot water or heating?

I am a new house owner and are having issues with the central heating and hot water. As of sunday, it went offline without warning. The gas flame in the boiler is fine, but the tank in the airing cupboard is cold, I believe it is a combi boiler? The panel in the bathroom states the heating and/or hot water go on-but they haven't. It's also started humming softly, when the panel is on but there is no sign of the water heating.

If we put the immersion on we get a limited amount of hot water, so the tank does heat up-it just won't run without it. Could you please advise as to what kind of problem this is? We have no idea whether this should be on/off-we know the boiler works-heats up-so it doesn't need replacing but we are unsure what's going on.

We have a plumber coming to look at it, but any ideas/advice would be much appreciated. Thanks