There's a couple assumptions that you have here that mostly have to do with recent phenomena and don't relate well to how industrial-revolution governments work. Since you mentioned the UK and it has a fair bit of information available, we'll use them as the primary example.

First, governments simply don't spend as much. Modern developed economies tend to take 50 to 30 percent of GDP in income; by contrast, from 1700 to 1850, British taxes as a share of national income (not quite GDP, but similar enough here) ranged from 9 to 13 percent, except for spiking up to 24% during the Napoleonic wars.

Second, income tax is not the most important tax in these years, compared to customs and excise tax: import fees, sales taxes on alcohol, sugar, tea, salt, candles, windows, etc etc. For the aforementioned period, these make up 59 to 72 percent of British government revenue.

Third, debt financing. Obviously, this is perfectly modern, but essentially ever war Britain fought was financed by selling bonds internally to the gentry and industrialists. If you've ever read Jane Austen, most of these characters are getting their yearly income not through their estates but by investment in British government bonds, particularly an old kind of bond called the consol, which was a perpetual bond: if you paid the government £100, each year the government paid you back (during this time period) 3%, forever, until they bought the consol back from you at the same price. This and other debt meant that most of the Napoleonic Wars were possible, making up £505 million of the £831 million cost of said wars.