There is no evidence that the British government wanted the war, and plenty that they did not.
Well done, Keith, to write "the British government" as opposed to "Britain". Of course, they would have preferred the other side to cave in, but they (and their newly won French allies) gambled with the risk.
If they wanted war, they might be expected to provide an army capable of fighting it.
Quite the opposite: since everybody had known for years in advance that war was imminent and the British government had committed itself to join, they would have to prepare for it even if they did not want it. The army was correctly judged sufficient for victory; the loss of life was part of the calculation, as Teribus described. Same in France.
And yes, some politicians in all countries had analyzed the situation of 1914 correctly, and already knew that the idea of a quick war was a miscalculation, but it seemed too late to stop it, for two reasons:
Others did want the war even if it lasted as long and took as many lives as it did (think of arms manufacturers, military leaders etc.),
On both sides, the propaganda had done its effect and could not be undone without loss of face. (Note that this is not just my guess, but statements to that effect are reported from France and Germany alike; your "distraught" Ministers point to the same direction.)
I do not know who exactly is to blame, for warmongering or for sleepwalking, and I do not consider this the most important question now that the conflict is over.
The main lesson to learn for us is to resist all nationalist propaganda. The second lesson is that "we did not want that war" is only credible if backed by an evidently fair offer of peace.