Process results from two different @wire methods after the @wires are finished

Option 1: use wire handlers. Instead of writing an Attribute name, you can write a function, and store the data in attributes, then call a common function when both results are available.

---

Example

attr1;
attr2;
@wire(getAttr1, {})
attr1result({data,error}) {
  if(data) {
    this.attr1 = data;
    if(this.attr1 && this.attr2) {
      this.doProcessing();
    }
  } // ... errors ... //
}
@wire(getAttr2, {})
attr2result({data,error}) {
  if(data) {
    this.attr2 = data;
    if(this.attr1 && this.attr2) {
      this.doProcessing();
    }
  } // ... errors ... //
}
doProcessing() {
  // do something with attr1 and attr2 //
}

---

Option 2: Use a promise chain. When calling them in order, keep in mind that this will cause a minor performance penalty.

---

connectedCallback() {
  getAttr1({}).then(result => {
    this.attr1 = result;
    return getAttr2({})
  }).then(result => {
    this.attr2 = result;
  }).finally(()=>this.doProcessing());
}

We can chain wire services by using properties that are both dynamic and reactive by using it as $properties and also decorating it with track.

In the below example I have chained getObjectInfo and getPicklistValues wire services in a manner in which I leverage the data from getObjectInfo into getPicklistValues

import {
    LightningElement,
    api,
    wire,
    track
} from 'lwc';
import {
    getObjectInfo
} from 'lightning/uiObjectInfoApi';
import {
    getPicklistValues
} from 'lightning/uiObjectInfoApi';

export default class PocPicklistFilters extends LightningElement {

    // Flexipage provides recordId and objectApiName
    // to be passed from calling component
    @api objectApiName = 'Account';
    @api picklistField = 'Account.Industry';
    defaultRecordTypeId; //Master  Record Type Id
    @track options = [];


    @wire(getObjectInfo, {
        objectApiName: '$objectApiName'
    })
    wiredObjectInfo({
        error,
        data
    }) {
        if (data) {
            console.log('ObjectInfo' + JSON.stringify(data));
            this.record = data.recordTypeInfos;
            this.defaultRecordTypeId = data.defaultRecordTypeId;
            this.error = undefined;
        } else if (error) {
            this.error = error;
            this.record = undefined;
        }
    }

    @wire(getPicklistValues, {
        recordTypeId: '$defaultRecordTypeId',
        fieldApiName: '$picklistField'
    })
    wiredPicklistInfo({
        error,
        data
    }) {
        if (data) {
            console.log('picklist' + JSON.stringify(data.values));
            this.processPicklistRawJSON(data.values);
        } else if (error) {
            console.log(JSON.stringify(error));
        }
    }

    processPicklistRawJSON(_rawData) {
        this.options = _rawData.map(row => {
            return {
                label: row.label,
                value: row.value
            };
        });
        console.log('picklistprocessed' + JSON.stringify(this.options));
    }
}