Okay, so there are two parts to this - the first part is an issue with the translation tables, and the second part is an issue with NeoLemmix itself.
The NeoLemmix part of this is fixed in commit 97079B3.
All translation tables currently included with NeoLemmix are fixed in commit A0E0396. To fix any custom ones - you need to go through it, find the $BACKGROUND entries, and change it so that the first one's INDEX is 0, the second is 1, etc. The fixed table, together with NeoLemmix V12.6.0 stable (or an experimental released later than this post), should be able to open the levels with the correct backgrounds in place.