You're confusing two things here - the fiscal deficit and the national debt.
If government spending is higher than government income (i.e. taxes and other revenues) then the government has to borrow to finance that gap. So debt will increase by the amount of the deficit (and should decrease if there's a surplus). We've only shown a surplus in 5 out of the last 40 years, including 1999-2001, so government debt will always be increasing. The real question is around the level of debt compared to the growth in the economy and how that impacts the servicing of that debt.
What's happened in the last 10 years or so is that Gordon Brown, as chancellor, did quite a good job of bringing down debt as a percentage of GDP, by ensuring that spending didn't run ahead of income. By 2002 it was down to around 37% of GDP, from a peak of 50% in 1996 as he'd been recording a surplus and paying debt down.
In 2002 he went on a bit of a splurge and started building up a deficit, when there was an argument that, with the strength of the economy, there should have been at least a balanced budget if not a surplus and debt should have been reduced further but debt as a % of GDP started to creep up to around 43% in 2007, when it really should have been not much more than 30%.
In 2008, government spending started to increase dramatically, as the liquidity crisis started to take effect and absolutely ballooned the following two years. By 2010, government debt had increased to nearly 80% of GDP as we were spending far, far, more than we earned as a country.
Since then, Osborne has been trying to reduce the gap between what we spend and what we earn. That's had some limited success and debt as a % of GDP is starting to level off at around 90%. The economic argument is that cuts have to be delicately managed so they don't constrain recovery, as it's recovery that drives up GDP and therefore public revenues. As demand grows, then consumption grows, production grows, income and profits grow and higher personal and business taxes increase government revenues. Some feel that the cuts have been too severe but they are certainly having an impact on the debt/deficit picture. That's at a cost to the ordinary person though.
But that's not happening this time, as I've explained before, due to growth in production not feeding into growth in personal, disposable incomes and this is limiting demand. That latter growth is needed to drive a sustainable recovery. Current projections are that the deficit should be cleared in the next 3 years and we should start recording a surplus by the end of 2018 and that government borrowing (which is not quite the same as the deficit) should cease by the end of 2019/2020. That's how they can talk about tax cuts by that time. The real test will be whether this is achieved or even achievable without a growth in incomes.