'Bolshie' did come from 'Bolshevik'. How it came to be applied to birds may not be demonstrable, but ...
My research suggests that 'Bolshie', a diminutive of 'Bolshevik', was first applied to Bolsheviks themselves, individually and in the plural as a group. As a diminutive, a linguistic mechanism, 'Bolshie' may be applied both affectionately and, to lessen a perhaps unacknowledged fear or anxiety of the thing diminished, contemptuously or otherwise approbatively.
The first appearance of 'Bolshie' in Google Books appears to be this from the Spectator in 1918:
HAVELOCK WILSON'S APPEAL. [To the Editor of the "Spectator."] Sir, — The General Election appears to be imminent and patriotic Labour must mobilize for the fray. The Bolshie Bosses control the Labour Party machinery and political funds ...
This apparently contemptuous reference appeared in the Spectator around the time of the end of World War I. During that war, the March 1917 collapse and replacement of the Russian government with the Russian Provisional Government and the Petrograd Soviet, followed by the Bolshevik revolution and armistice with the Central Powers in December of 1917, was a telling victory for Germany.
Notably, it is not much later than the 1918 appearance of 'Bolshie' in the Spectator that 'Bolshy-bird' appears in a poem in a 1919 volume of Punch:
... it should contrive
To keep its pinions on the flap,
And by a tour de force survive
This devastating handicap,
Yet are there perils in the skies
Whereon we blandly shut our eyes,
But which are bound to be incurred,
And, notably, the Bolshy-bird.
This was, I surmise, an extended metaphor with direct reference to the Bolsheviks. It seems likely that the phrase 'bolshy-bird' was, in part, appealing because of the b-b alliteration.
Next, in the 1920 Mr. Punch's History of the Great War (Cassell, Limited, 1920 - World War, 1914-1918), we find this:
Here again, the 'bolshy-bird' appears metaphorically, this time as a "most obscene brand of vulture". The force of the comparison draws on the notion of the Bolsheviks preying on the ruins of the Russian Empire.
By 1940, 'bolshy' was established slang meaning "a person of violent and disagreeable habits":
1940 ‘G. Orwell’ in Horizon Mar. 187 If the Russian Revolution is anywhere referred to it will be indirectly, in the word ‘Bolshy’ (meaning a person of violent disagreeable habits).
["Bolshy | Bolshie, n. and adj.". OED Online. December 2015. Oxford University Press. http://www.oed.com/view/Entry/21134?redirectedFrom=bolshie (accessed December 11, 2015).]
The "violent and disagreeable" slang meaning was subsequently applied directly (as opposed to the earlier metaphorical use) to some birds, the observed behavior of which around bird feeders and in the wild was "violent and disagreeable".
Blackcaps, which chase other birds away from feeders, frequently earn the nickname 'bolshie'.
Blackcaps have a reputation for being particularly bolshie, frequently shooing other birds away from feeders, but is this a fair assessment?
(From Fatbirder - linking birders worldwide..., in "Sightings of Bolshie Garden Bird", 2013.)
Bolshie bird? Blackcaps have a growing reputation for being grumpy at bird feeders, shooing off other birds that come to feed.
(From BTO: Looking out for birds, "Key questions and results", undated.)
Any bird that displays similar behavior, however, is likely to be described as 'bolshie', with affectionate as often as disparaging force:
(From an article in the Daily Express, 2002.)
So, when the falcon or raptors in general are described as a 'bolshie birds', the meaning is the slang meaning: they are violent and disagreeable, like the Bolsheviks.