backend.py

from wagtail.wagtailsearch.backends.elasticsearch import ElasticsearchSearchBackend, ElasticsearchSearchQuery


class ElasticSearchQueryWithFuzziness(ElasticsearchSearchQuery):
    def get_inner_query(self):
        if self.query_string is not None:
            fields = self.fields or ['_all', '_partials']

            if len(fields) == 1:
                query = {
                    'match': {
                        fields[0]: {
                            'query': self.query_string,
                            'operator': self.operator,
                            'fuzziness': 'AUTO',
                        }
                    }
                }
            else:
                query = {
                    'multi_match': {
                        'query': self.query_string,
                        'fields': fields,
                        'operator': self.operator,
                        'fuzziness': 'AUTO',
                    }
                }
        else:
            query = {
                'match_all': {}
            }

        return query


class CustomElasticSearch(ElasticsearchSearchBackend):
    search_query_class = ElasticSearchQueryWithFuzziness


SearchBackend = CustomElasticSearch

Here's a version that works with Wagtail 2.4 & Elasticsearch6. I have only tested this briefly.

from wagtail.search.backends.elasticsearch6 import Elasticsearch6SearchBackend, Elasticsearch6SearchQueryCompiler


class ElasticSearchQueryWithFuzziness(Elasticsearch6SearchQueryCompiler):
    def _compile_plaintext_query(self, query, fields, boost=1.0):
        match_query = {
            'query': query.query_string,
            'fuzziness': 'AUTO',
        }

        if query.operator != 'or':
            match_query['operator'] = query.operator

        if boost != 1.0:
            match_query['boost'] = boost

        if len(fields) == 1:
            return {
                'match': {
                    fields[0]: match_query
                }
            }
        else:
            match_query['fields'] = fields

            return {
                'multi_match': match_query
            }


class CustomSearchBackend(Elasticsearch6SearchBackend):
    query_compiler_class = ElasticSearchQueryWithFuzziness


SearchBackend = CustomSearchBackend

or perhaps alternatively:

class ElasticSearchQueryWithFuzziness(Elasticsearch6SearchQueryCompiler):
    def _compile_plaintext_query(self, *args, **kwargs):
        query = super()._compile_plaintext_query(*args, **kwargs)
        query_type = list(query.keys())[0]
        query[query_type]['fuzziness'] = 'AUTO'
        return query