Apparently they need to be in an enclosure as there is only single insulation on the join.
Those splices are meant for overhead lines, not where they can be touched.
That mess shouldn't be in the cabinet anyway.
They could have had a Combi previously, and have now had a replacement Combi so never had an immersion in the first place.
The OP right from the start said that there is NO immersion heater or anything like it.
There are certain questions that the Op hasn't replied to yet.
But even if there is an immersion tank being heated, can you explain why there would be an increase in consumption to previous ? The hot water would now be supplied from the combi.
A bit more information is needed.
Are we talking about upstairs /downstairs or all.
If only the first light in a string/loop is working, then it suggests a break in the wiring could be due to rodents ?
Have you checked other roses ?
Has there been a change in the way the house was heated, such as storage heaters previously ?
Has there been a change of meter ?
Is there an immersion tank at all ?
A picture of the consumer unit and meter, including showing the mcbs/rcbo circuits, maybe of help