You will need an action for a relookup such as using the script step or your ID field being updated.
Most of the time, the best approach for this is not to use a look up field. If possible, replace your lookup field with a reference to the actual related field from which you are currently looking up values.
This could be as simple as placing the related field on your layout in place of the current lookup field.
If you use the look up field in calculations, you can edit the calculation to refer to the related field instead, though this results in an unstored field and that may or may not work for you.
BTW, the Relookup didn't work for you because you placed the focus on the wrong field. If you click into the key field that defines the relationship you use in the lookup. ReLookup will then work for you.