They should operate an open submission policy though i.e. a bug report form and actually feed back to people.
Granted that in an ideal world this sort of incentive wouldn't be necessary, but if we lived in an ideal world, code wouldn't have bugs in it, too.
And another reason: If the government pays bounties for confirmed bug reports, a prospective reporter can have some confidence that they won't react to the report by taking punitive action, which might otherwise be a very real concern; HN periodically sees threads about employers and clients reacting extremely badly to unsolicited exploit reports, and the worst those can do stops short of imprisoning somebody.
Of course, all of this assumes a government not so sclerotic that it'll take an interest in fixing bugs without first having to be motivated by a horribly embarrassing public compromise, which means it's probably not going to work in the UK, or for that matter in the US either. But it's a nice thought, I suppose.
Bug bounty rewards are orders of magnitude cheaper than the damage the bugs can cause.
If they have a bug report form at all and employ people to screen and respond to bug reports, the bounty costs probably wont even add up to the cost of 1 additional employee
They're pissing away profit and investor's cash in private companies.
In public organisations, taxpayers don't get to choose if that money is spent or not.