Weird texture problem in HTML5

do not trust mobile/html5 preview.

most probably you are using a material type or node that is not compatible with html5, and your current one is packed incorrectly.
try checking output log for warnings about material compatibility errors. something similar to this:

MainFrameActions: Packaging (HTML5): UE4Editor-Cmd: [2016.04.26-19.58.17:556] 0]LogInit:Display: LogMaterial:Warning: Failed to compile Material /Engine/EditorMaterials/PreviewShadowIndicatorMaterial.PreviewShadowIndicatorMaterial for platform GLSL_ES2_WEBGL, Default Material will be used in game.

material editor will also tell you what cannot be used when you are in mobile renderer after clicking the apply button.

another possibility is that, you are clamping your texture. make sure that the texture’s tiling methods are “wrap” for both x and y.
Capture.JPG

this is how a clamped texture shows in material editor. however, it will show tiled on screen.