You hvae my sympathy Blackbirdboxer - I have the same problem with my hedge - it has nesting blackbirds, blue tits, sparrows and goldfinch as far as I can determine, so I am unable to trim it.
And, worse than that, there's a Starling's nest in my house downpipe which now overflows every time it rains.
Moving an empty nest would be fine I would have thought - I don't think the law would be enacted for such a case, especially with your obvious caring nature.
I imagine that lots of (human) activity in the shed would also deter a second brood in that locality. You could try asking the RSPB for advice on their website