How much did Great Britain, Russia and France spend in 1890 and in 1914?

There are two questions you could be asking here, but it's simple enough to answer both.

The first is how much they spent in general, which is by definition their GDP.

I've used the work of Paul Bairoch, then converted his figures from 1960 dollars to 2015 dollars using the BLS CPI inflation calculator.

I couldn't find 1914 exactly, but if you use 1913 and assume the same average growth rate you can estimate 1914 fairly closely.

Great Britain, 1890: $235.5 billion
France, 1890: $158.1 billion
Russia, 1890: $169.4 billion

Great Britain, 1913: $352.6 billion
France, 1913: $219.2 billion
Russia, 1913: $419.4 billion

Great Britain, 1914 estimate: $359.8 billion
France, 1914 estimate: $222.3 billion
Russia, 1914 estimate: $427.5 billion

The general sense you can get is that while Britain started the biggest and they were all close together, by the end Russia was the biggest and France fell far behind.

But since you asked about specific years that seem to relate to important wars, the other question you might have meant to ask is how much each country spent on the military. I have those figures as well. They were given in 1934 dollars, which I converted to today's money as well.

Great Britain, 1890: $3.05 billion
France, 1890: $3.35 billion
Russia, 1890: $6.47 billion

Great Britain, 1914: $6.62 billion
France, 1914: $6.43 billion
Russia, 1914: $7.70 billion

As you can see, they all increased their military spending quite a bit, but Russia started with the highest by far and France and Great Britain caught up.

A word of caution: A lot of these figures come with considerable uncertainty. Different historians use different measures and different estimates of inflation; I've reported them to a couple of decimal places but that's probably not actually realistic. The general trend, however, is known: Both overall GDP and military spending increased substantially during this period, but contrary to popular belief of a long-term trend of "militarization", GDP actually increased a good deal faster than military spending. It was only in the immediate lead-up to World War 1, starting around 1910, that military spending really began to rise much faster than other spending.