Forgive the broad and simplistic wording of the question, but I'm still learning.
I understand that Keynesian economics, in simple terms, requires saving and restricting public spending when times are good and then borrowing and spending money when times are bad, to apparently make up for a shortfall in spending and growth. The idea seems to rely on budget surpluses and debt repayments in good times, if I've understood things correctly?
What I understand of Gordon Brown's approach was that money was increasingly taxed, borrowed and spent during the good times and accelerated in the bad (and it would seem that nothing has fundamentally changed with the new government).
I understand that Austrians disagree with the Keynesian model, but do they think it's more preferable than the model that Gordon Brown followed?
What you're describing there is Fiscal Keynesianism. That hasn't existed in mainstream economics for a long time. When you say "Keynesian" nowadays you're actually referring to the New Keynesian school, which is closer to an extension of monetarism than anything in the General Theory. In New Keynesian economics, aggregate demand is entirely managed by the central bank, usually by way of an inflation (or other) target. Assuming a binding target, any increase/decrease in government spending (including counter-cyclical spending) will be entirely offset by tightening/easing of monetary policy. Ie, government spending usually won't work and will just raise the future tax burden.
In the liquidity trap, the job of the central bank becomes very difficult. There was question over whether monetary policy has any effect, and so people like Krugman advocated counter-cyclical fiscal spending (which wouldn't be offset by tightening monetary policy). Now it pretty quickly became clear that fiscal stimulus adequate to close the output gap isn't politically feasible, and it became more apparent that the central bank wasn't "out of bullets". However, if a central bank has allowed nominal income to fall far below trend, and is not taking appropriate action to restore it, this is effectively tight monetary policy and fiscal austerity (which is desirable) will not be offset by expansionary monetary policy.
Most modern Keynesians would prefer that the central bank do their job (so far the Riksbank is the only central bank that has managed to restore nominal income close to its pre-GFC trend), rather than have the government engage in stimulus. I think a Keynesian would prefer that Gordon Brown direct the chancellor of the exchequer to give the Bank of England a higher inflation target or a NGDP target for a while.