For BE, it is two words. WR dictionary (Concise Oxford), fairy tale:▶noun a fairy story. ■ [as modifier] typical of a fairy story; magical or idealized: a fairy-tale romance.
So fairytale-like would be fine in AE, but for BE, considering the "fairy-tale romance" examples above, the form would seem to be fairy-tale-like. Another option might be to leave fairy tale as two words and connect like with an en dash:fairy tale–like.