Europe didn't take a break from collecting measurements during the world wars. On the contrary, weather forecasting is essential for planning and executing naval and air operations, and the various powers went to great lengths to maintain weather stations in even remote and inhospitable areas towards this purpose. For example, wiki's got a nice graph of Norwegian temperature collections on Svalbard back to 1912, showing a distinct warming trend over the period, particularly in recent decades.
A lot of weather data collection in the first half of the century was performed at sea by specially equipped ships. I sailed recently on a windjammer whose original purpose was meteorological measurements in the north Atlantic. The American Institute of Physics has an interesting article on climate change with a great deal of history included, in particular the following: It may have been the press reports of warming that stimulated an English engineer, Guy Stewart Callendar, to take up climate study as an amateur enthusiast.
He undertook a thorough and systematic effort to look for historical changes in the average global temperature. One 19th-century German had made an attempt at this in seeking a connection with sunspot cycles. Otherwise, if anyone else had thought about it, they had probably been discouraged by the scattered and irregular character of the weather records, plus the common assumption that the average climate scarcely changed over the span of a century.
But meteorologists around the world had meticulously compiled weather records, and Callendar drew upon that massive international effort. After countless hours of sorting out data and penciling sums, he announced that the temperature had definitely risen between 1890 and 1935, all around the world, by close to half a degree Celsius (0.5°C, equal to 0.
9°F). There was plenty of data being taken as a result of the surge in interest in the late 19th century in geophysics and study of the earth's climate, but the effort wasn't necessarily uniform or well disseminated. Fortunately, there's still a wealth of data available, and today modern computers let us perform large-scale numerical analysis on large databases rather than "penciling sums".
The methodology for taking measurements Europe didn't take a break from collecting measurements during the world wars.