Tilekit exports to JSON by default -- this can be read by any JSON library (of which any language should have readily available), from there it should be trivial to get the resultant data into your engine's tilemap structure; the objects should be equally trivial to load. The structure of the exported format is detailed in the manual.

In addition to this, Tilekit can also export to plain CSV, as well as Tiled's TMX format -- the latter can be converted by Tiled to any of its own export formats.

I can upload a map exported to both JSON and TMX if you want to experiment with loading before purchasing!