When Battle results has a new version, the garage panel says so and its Download button opens this page. Close the game, get Battle results again from its page on the Mod Hub and extract it over the old files. A mod's file is named after its version: if it is left with two files, delete the older one. Only that mod changes; your settings stay in mods\configs\app.gunmarks.
Battle results needs no Gunmarks Core. app.gunmarks.core_*.wotmod is for the mods of 0.1.1 and earlier, which do: once every mod of yours is newer than that, delete it. The settings page says when.
After a World of Tanks update the game uses a new mods folder, named after its version: wait for the mod’s release for it.