In QGIS you can use the Reclassify by table
tool. Go to the Processing menu toolbar and select Toolbox then, go to Raster Analysis > Reclassify by table. Or just search for the tool. At that point it is fairly simple to use, just define your raster and output then click on the ellipse (...) next to the Reclassification table box. This will bring up an interactive GUI that lets you define the reclassification parameters.
The difficult part will be in defining the reclassification itself. The classification domains between NLCD and LCCS are quite different. It may very well be easier to classify both products into a common, newly defined classification domain that more consistently represents both classifications. Below are the two classification domains. As you can see LCSS classes are more general and you are wanting to collapse them into a system that has more refined classes. Collapsing NLCD into LCSS makes much more sense.
UN LCCS classes
(1) Cultivated and Managed Terrestrial Areas; (2) Natural and Semi-Natural Terrestrial Vegetation
(3) Cultivated Aquatic or Regularly Flooded Areas; (4) Natural and Semi-Natural Aquatic or Regularly Flooded Vegetation
(5) Artificial Surfaces and Associated Areas; (6) Bare Areas
(7) Artificial Waterbodies, Snow and Ice; (8) Natural Waterbodies, Snow and Ice.
NLCD classes
(11) Open Water; (12) Perennial Ice/Snow
(21) Developed, Open Space; (22) Developed, Low Intensity
(23) Developed, Medium Intensity; (24) Developed, High Intensity
(30) Transitional; (31) Barren Land (Rock/Sand/Clay)
(41) Deciduous Forest; (42) Evergreen Forest
(43) Mixed Forest; (51) Dwarf Scrub
(52) Scrub/Shrub; (71) Grassland/Herbaceous
(72) Sedge/Herbaceuous; (73) Lichens
(74) Moss; (81) Pasture/Hay
(82) Cultivated Crops; (90) Woody Wetlands
(95) Emergent Herbaceous Wetlands