Fair enough…
from what I can make of this:
Under the uncodified British constitution, executive authority lies with the monarch, although this authority is exercised only by, or on the advice of, the Prime Minister and the Cabinet.[5] The Cabinet members advise the monarch as members of the Privy Council.
Any claims that the Queen is just a figurehead is bollocks, especially given the secrecy surrounding the Privy Council, not to mention this link mentioning the attempts by the government to keep the Queen’s involvement hidden.
With regards to the original questions:
a) Just how many armies and intelligence services does the Queen have authority over?
b) Does this mean that Tony Blair isn’t the only one to blame for the Iraq War?
a) The Queen certainly has authority over the Armies and intelligence services of not only the UK, but also the Commonwealth Realms:
The monarch is the head of state of each of the Commonwealth realms, charged with issuing executive orders, commanding the military forces, and creating and administering laws.[37] However, each country now operates under the Westminster system of parliamentary democracy and the concept of responsible government, meaning that the monarch exercises her powers only on the advice of her Crown ministers, who are usually drawn from, and thus responsible to, the elected chamber of the relevant parliament.
Just how many Armies and Intelligence/security services come under her authority in all, I’m not entirely sure~ I doubt every last one of the Commonwealth Realms and British Overseas Territories have their own Armies and intelligence services, but those that lack their own will probably have local coverage on the Queen’s behalf.
b) It certainly doesn’t seem that Tony Blair could’ve authorized war on Iraq without the Queen’s consent, especially in light of this:
The Queen also vetoed entirely a private member’s Bill, the Military Actions Against Iraq (Parliamentary Approval) Bill 1999, that would have transfered the power to authorise military strikes against Iraq from the monarch to Parliament
Whether that point will be acknowledged in the Chilcot report is yet to be seen, however it is worthy of note that John Chilcot is himself on the Privy Council.