Downstairs flat using our electricity. What are our rights?

dogshome

Don't think it's a case of Electricity theft, more likely that in earlier days the garage was for use of of the Maisonette and in some long forgotten deal, it's use was changed to the occupiers of the flat

If this is the case it's going to be difficult to identify who was responsible, and even more difficult to get recompense

Probably best to mark up the switches on the consumer unit, keep them switched off with a note on the box as to why they should not be switched on, and inform the occupiers of the flat of the situation in WRITING 

Ditzy_Mitzy

A maisonette is a self contained thing thus, I imagine, the maisonette's fuse box is located within it; rather than being sited in a communal area.  Owing to this, the only possible causes for this situation are a deliberate action on the part of a previous occupant of the maisonette, or the tenant of the other flat gaining access without permission and tapping into the fuse box.  The second notion is unlikely, therefore responsibility for the electrical discrepancy is likely to lie with a previous occupant of the maisonette.  
It's more than likely that, in times past, the garage was part of the maisonette's demise, rather than the flat's.  At that time, therefore, it stood to reason for the garage to be wired up to the maisonette's meter.  The custody of the garage has since swapped, for some reason, but whoever dealt with the swapping either didn't realise that the wiring needed altering, or wasn't sufficiently bothered.  Perhaps there was a casual arrangement put in place for the flat occupant to pay the maisonette occupant a small fee?  Anything could have happened, but the point is that it's not really relevant now. 
So the question is: if you want your money back, who are you prepared to sue?  It's not the fault of the occupant or landlord of the downstairs flat, not really.  The rogue electrical connection is on your property, so you can't claim that either party knows, or ought to know, that it exists until the point when you specifically alert them to the problem. You might argue that the tenant, presumably the bill payer, should have noticed that his bills were lower than expected, but again that's difficult.  His bills have always been the same, as you have been powering the garage throughout his tenancy.  
You could make the same argument that, in fact, it was your own fault for either not noticing that the usage was too high or, as you suggest, becoming aware of the problem but not consulting an electrician.  There are things you, with access to the meter and fuse box could have done: turning everything off and checking for current draw, for instance. 
The previous occupants: were they negligent for not telling the new owner of the maisonette that it powered the garage?  Maybe, but then you'd need to look at when the change in custody of the garage, if that is the case, happened and the circumstances.  Do you even know who they were?
Finally: is there any reciprocal benefit?  You mentioned the neighbour using a lawnmower.  Is that on a communal area or a private garden?  If it's the former, then you derive some benefit from the mowing being done.  
This is extremely complicated when looked at retrospectively and we haven't even got into what you estimate the excess usage to be, and how you'd have a prayer when it comes to proving it, should the party you attempt to recover money from wish to dispute the matter. 
It's probably easiest, all said and done, to sort this out now and forget about what happened in the past.  You could, either, re-route the garage's electricity supply via the flat's meter or come to an agreement for the flat's occupant to pay you something towards the electricity.  

dogshome

Further thought - Is your rented maisonette and the flat below owned by the same landlord ? - In which case put the situation in WRITING to Him/Her/Company underlining that the present set-up has not only financially dis-advantaged you, but is dangerous as has already been proven by an electrician working in the garage blowing your electrics, when as a professional he would already have shut off what he thought was all power from the flats own consumer unit, pus the fact that if a fault occurs in an appliance/plug/wire or socket, in the garage, the occupiers have no means of killing the power supply

Till the situation is changed - KEEP THE SWITCHES YOUR CONSUMER UNIT SWITCHED OFF