If one isn’t islamically married to their fiancé, but fears that one may fall into lust and sin, is it then wajib to do nikkah immediately, even if the families want to wait some months and are against having nikkah done immediately?

Yes In your specific case it’s advisable you do a conditional temporary nikkah as in Muta where the condition will be not to consummate the marriage till the Permenant one takes place.