It's not exactly easy to build all pieces, but here you go. I think that someone can help only if he/she tried something similar before. I'm sure that if the question would have been for VisualForce, many developers tried this type of page before. My question is specifically on these "new" components: <lightning:tab>, <lightning:datatable>, not about obtaining the same functionality using some other (more simple) aura components.
Once again: all data is displayed fine, sorting works fine, column resizing DOESN'T work correct (except for the table in the first tab, where is ok).
Thank you for trying to help!


The server method is working fine returning:
@AuraEnabled
    public static Map<String, Object> getData()
    {
        Map<string, Object> returnValue = new Map<string, Object>();
        ///data is selected from custom object and loaded into returnValue
        return returnValue;
    } 
The component code is:
==================
<aura:component implements="force:appHostable,flexipage:availableForAllPageTypes" controller="LTController_TyDash" >
    <aura:handler name="init" value="{!this}" action="{!c.doInit}"/>
    <aura:attribute name="avgColumns" type="List"/>
    <aura:attribute name="stColumns" type="List"/>
    <aura:attribute name="accAvg" type="Object"/>
    <aura:attribute name="accNew" type="Object"/>
    <aura:attribute name="accWor" type="Object"/>
    <aura:attribute name="accDis" type="Object"/>
    
    
    <!-- PAGE HEADER -->
    <div class="slds-page-header slds-page-header_object-home">
        <lightning:layout >
            <lightning:layoutItem >
                <lightning:icon iconName="standard:business_hours" />
            </lightning:layoutItem>
            <lightning:layoutItem class="slds-m-left_small">
                <p class="slds-text-title_caps slds-line-height_reset">XXXX</p>
                <h1 class="slds-page-header__title slds-p-right_x-small">Activities Dashboard</h1>
            </lightning:layoutItem>
        </lightning:layout>
    </div>        
    
    <div class="slds-grid slds-wrap slds-grid_pull-padded">
          <div class="slds-p-horizontal_small slds-size_1-of-1">
            <lightning:tabset variant="scoped">
                <lightning:tab label="PROSPECT STAGES (AVERAGES)" id="avg" >                   
                    <lightning:datatable data="{! v.accAvg}" aura:id="accAvg"  
                       columns="{! v.avgColumns}" 
                       keyField="name"                       
                       onsort="{! c.updateColumnSorting}"                                 
                       sortedBy="name"
                       sortedDirection="asc"
                       defaultSortDirection="asc"
                       hideCheckboxColumn="true"/>                           
                </lightning:tab>
        <lightning:tab label="SUSPECTS - NET NEW" id="new" >
            <lightning:datatable data="{! v.accNew}" aura:id="accNew"  
                       columns="{! v.stColumns}" 
                       keyField="account"
                       onsort="{! c.updateColumnSorting}"               
                       sortedBy="owner"
                       sortedDirection="asc"
                       defaultSortDirection="asc"
                       hideCheckboxColumn="true"/>    
        </lightning:tab>
        <lightning:tab label="SUSPECTS - WORKING" id="wor" >
            <lightning:datatable data="{! v.accWor}" aura:id="accWor"  
                       columns="{! v.stColumns}" 
                       keyField="account"
                       onsort="{! c.updateColumnSorting}"               
                       sortedBy="owner"
                       sortedDirection="asc"
                       defaultSortDirection="asc"
                       hideCheckboxColumn="true"/>
        </lightning:tab>
        <lightning:tab label="PROSPECTS - MI SET (DISCOVERY)" id="dis" >
            <lightning:datatable data="{! v.accDis}" aura:id="accDis"  
                       columns="{! v.stColumns}" 
                       keyField="account"
                       onsort="{! c.updateColumnSorting}"               
                       sortedBy="owner"
                       sortedDirection="asc"
                       defaultSortDirection="asc"
                       hideCheckboxColumn="true"/>
        </lightning:tab>
    </lightning:tabset> 
        </div>
    </div>
</aura:component> 

Controller code:
============
({
    doInit : function(component, event, helper) {
        helper.loadData(component);
    },
    updateColumnSorting : function(component, event, helper) {    
        
        console.info('>>event ', event.getSource().getLocalId());
        var tableId = event.getSource().getLocalId();
        
        var fieldName = event.getParam('fieldName');
        var sortDirection = event.getParam('sortDirection');        
        var avgT = component.find(tableId);
        avgT.set("v.sortedBy", fieldName);
        avgT.set("v.sortedDirection", sortDirection);
        helper.sortData(component, tableId, fieldName, sortDirection);
    }
})

Helper code:
===========
({
    loadData : function(component) {
        var action = component.get("c.getData");        
        action.setCallback(this, function (result) {
            var state = result.getState();
            if (state !== "SUCCESS") {
                console.info('>> getting daa was not successful:', result);
                return;
            }
            var data = result.getReturnValue();                        
            console.info('>> oninit data:', data);
            
            component.set("v.avgColumns", [
                    {label: "SALES REP NAME", fieldName:"name", type:"text", sortable: "true"},
                    {label: "#DAYS in NET NEW (average)", fieldName:"avgNew", type:"number", sortable: "true"},
                    {label: "#DAYS in WORKING (average)", fieldName:"avgWor", type:"number", sortable: "true"},
                    {label: "#DAYS in MI SET (average)", fieldName:"avgDis", type:"number", sortable: "true"},
                    {label: "#DAYS in PIPELINE (average)", fieldName:"avgPip", type:"number", sortable: "true"},
                   
                ]);
            component.set("v.stColumns", [
                    {label: "ACCOUNT NAME", fieldName:"account", type:"text", sortable: "true"},
                    {label: "Account OWNER", fieldName:"owner", type:"text", sortable: "true"},
                    {label: "#ACTIVITIES SINCE ASSIGNED to OWNER", fieldName:"cnt", type:"number", sortable: "true"},
                    {label: "#ACTIVITIES LAST 14 DAYS", fieldName:"cnt14", type:"number", sortable: "true"},
                    {label: "ACCOUNT STAGE", fieldName:"stage", type:"text", sortable: "true"},
                    {label: "#DAYS IN CURRENT STAGE", fieldName:"cntcrt", type:"number", sortable: "true"}
                ]);
                
            component.set("v.accAvg", data.t1);
            component.set("v.accNew", data.tn);
            component.set("v.accWor", data.tw);
            component.set("v.accDis", data.td);
        
        });
        $A.enqueueAction(action);
    },
    sortData: function (component, tableId, fieldName, sortDirection) {        
        var data = component.get("v." + tableId);
        var reverse = sortDirection !== 'asc';
        //sorts the rows based on the column header that's clicked
        data.sort(this.sortBy(fieldName, reverse))
        component.set("v." + tableId, data);
    },
    sortBy: function (field, reverse, primer) {
        var key = primer ?
            function(x) {return primer(x[field])} :
            function(x) {return x[field]};
        //checks if the two rows should switch places
        reverse = !reverse ? 1 : -1;
        return function (a, b) {
            return a = key(a), b = key(b), reverse * ((a > b) - (b > a));
        }
    }
})