こんな感じで出来るはず。

camelConverter.js

class CamelConverter {
  static convert(obj)  {
    return Object.keys(obj).reduce((acc, key) => {
      // 値がobjectの場合は再帰的にconvertを実行
      if(Object.prototype.toString.call(obj[key]) === '[object Object]') {
        acc[this.toCamel(key)] = this.convert(obj[key]);
        return acc
      }
      acc[this.toCamel(key)] = obj[key];
      return acc
    }, {});
  };

  // snake_caseの単語をlowerCamelに変換する
  static toCamel(str) {
    return str.replace(/_(\w)/g, (_match, capture) => {
      return capture.toUpperCase()
    });
  };
}

export default CamelConverter

テストはこちら（Jest）

camelConverter.test.js


import CamelConverter from './camelConverter'

describe('CamelConverter', () => {

  describe('#convert', () => {
    test('Key名をキャメルケースに変換する', () => {
      const snakeCase = {
        id: 1,
        property_string: 'string',
        property_int: 1,
        property_boolean: false,
        property_object: {
          inner_property_string: 'string',
          inner_property_int: 1,
          inner_property_boolean: false,
        }
      };
      expect(CamelConverter.convert(snakeCase)).toStrictEqual({
        id: 1,
        propertyString: 'string',
        propertyInt: 1,
        propertyBoolean: false,
        propertyObject: {
          innerPropertyString: 'string',
          innerPropertyInt: 1,
          innerPropertyBoolean: false,
        }
      })
    });
  })
})