There are some basic problems

• You have a unfiltered query(Highlighted in RED), Make sure you add proper filter there i.e a proper WHERE statement
• The query inside the execute method follow the basic governor limits like 50000 query rows etc. Only the query in the start statement(Highlighted in BLUE) can fetch upto 50million records

global with sharing class Populate_Corp_List_Assignments_Batch implements Database.Batchable < sObject > {

    global String Query;

    public List < Corporate_Account_List_Assignment__c > calist {
        set;
        get;
    }


    //given a AccountId and List<AccountTeamMember >, it returns the record having AccountId=AccoundId
    public List < AccountTeamMember > findAccountTeamMember(Id accountId, Map < Id, List < AccountTeamMember >> acc_tmListMap, String teamMemberRole) {
        if (acc_tmListMap == null || acc_tmListMap.size() == 0 || accountId == null) {
            return null;
        }

        integer count = 0;
        List < AccountTeamMember > atmList = acc_tmListMap.get(accountId);
        List < AccountTeamMember > atmRetList = new List < AccountTeamMember > ();

        if (atmList != null) {
            for (AccountTeamMember am: atmList) {
                if (am.TeamMemberRole == teamMemberRole) {
                    atmRetList.add(am);
                    count++;
                }
            }
        }
        if (count > 0) {
            return atmRetList;
        } else {
            return null;
        }
    }
    public Account findAccount(Id account_name, List < Account > accList) {
        if (accList == null || accList.size() == 0 || account_name == null) {
            return null;
        }
        for (Account am: accList) {
            if (am.Id == account_name) {
                return am;
            }
        }
        return null;
    }


    global Populate_Corp_List_Assignments_Batch(String q) {
        Query = q;
    }

    global Database.QueryLocator start(Database.BatchableContext BC) {

        return Database.getQueryLocator(query);
    }

    global void execute(Database.BatchableContext BC, List < sObject > scope) {


        //calist = [SELECT Account_Name__c,Corporate_Partnership_Director_1__c,Corporate_Partnership_Director_2__c,Corporate_Partnership_Director_3__c,Corporate_Partnership_Director_4__c,Corporate_Partnership_Director_5__c,Marketing_Solutions_1__c, Marketing_Solutions_2__c, Marketing_Solutions_3__c, Media_Services_1__c, Media_Services_2__c, Media_Services_3__c, Digital_Manager_1__c,Digital_Manager_2__c,Digital_Manager_3__c,Digital_Manager_4__c,Digital_Manager_5__c, Id,Name,Ultimate_Parent_Name__c FROM Corporate_Account_List_Assignment__c];     
        calist = [SELECT Account_Name__c, Corporate_Partnership_Director_1__c, Corporate_Partnership_Director_2__c, Corporate_Partnership_Director_3__c, Corporate_Partnership_Director_4__c, Corporate_Partnership_Director_5__c, Marketing_Solutions_1__c, Marketing_Solutions_2__c, Marketing_Solutions_3__c, Media_Services_1__c, Media_Services_2__c, Media_Services_3__c, Digital_Manager_1__c, Digital_Manager_2__c, Digital_Manager_3__c, Digital_Manager_4__c, Digital_Manager_5__c, Id, Name, Ultimate_Parent_Name__c FROM Corporate_Account_List_Assignment__c];
        //prepare a map of ids and account_name__c
        Map < Id, Id > ca_acc_map = new Map < Id, Id > ();

        for (Corporate_Account_List_Assignment__c ca: calist) {

            ca_acc_map.put(ca.Id, ca.Account_Name__c);

        }


        Map < Id, Account > ac2 = new Map < Id, Account > ([SELECT Name, ParentId, Id FROM Account where Id in : ca_acc_map.values()]);
        List < AccountTeamMember > atmcpd2 = [SELECT Id, TeamMemberRole, UserId, IsDeleted, AccountId FROM AccountTeamMember WHERE AccountId in : ac2.keyset() AND Isdeleted = FALSE];

        Map < Id, List < AccountTeamMember >> acc_tmListMap = new Map < Id, List < AccountTeamMember >> ();
        for (AccountTeamMember atm: atmcpd2) {
            List < AccountTeamMember > atmlist;
            if (acc_tmListMap.get(atm.AccountId) == null) {
                atmlist = new List < AccountTeamMember > ();

            } else {
                atmlist = acc_tmListMap.get(atm.AccountId);


            }

            atmList.add(atm);
            acc_tmListMap.put(atm.AccountId, atmList);

        }

        for (Corporate_Account_List_Assignment__c ca: calist) {
            ca.Corporate_Partnership_Director_1__c = null;
            ca.Corporate_Partnership_Director_2__c = null;
            ca.Corporate_Partnership_Director_3__c = null;
            ca.Corporate_Partnership_Director_4__c = null;
            ca.Corporate_Partnership_Director_5__c = null;

            ca.Digital_Manager_1__c = null;
            ca.Digital_Manager_2__c = null;
            ca.Digital_Manager_3__c = null;
            ca.Digital_Manager_4__c = null;
            ca.Digital_Manager_5__c = null;

            ca.Marketing_Solutions_1__c = null;
            ca.Marketing_Solutions_2__c = null;
            ca.Marketing_Solutions_3__c = null;

            ca.Media_Services_1__c = null;
            ca.Media_Services_2__c = null;
            ca.Media_Services_3__c = null;
            System.debug('******ID********' + ca.Id);


            //find atmcpd
            Account accX = ac2.get(ca.Account_Name__c);
            List < AccountTeamMember > atmcpd = findAccountTeamMember(accX.Id, acc_tmListMap, 'Corporate Partnership Director');
            List < AccountTeamMember > atmMS = findAccountTeamMember(accX.Id, acc_tmListMap, 'IMS');
            List < AccountTeamMember > atmMDS = findAccountTeamMember(accX.Id, acc_tmListMap, 'Media Services');
            List < AccountTeamMember > atmDM = findAccountTeamMember(accX.Id, acc_tmListMap, 'Digital Manager');


            if (atmcpd != null && atmcpd.size() > 0) {
                if (atmcpd.size() > 0)
                    ca.Corporate_Partnership_Director_1__c = atmcpd.get(0).UserId;

                if (atmcpd.size() > 1)
                    ca.Corporate_Partnership_Director_2__c = atmcpd.get(1).UserId;

                if (atmcpd.size() > 2)
                    ca.Corporate_Partnership_Director_3__c = atmcpd.get(2).UserId;

                if (atmcpd.size() > 3)
                    ca.Corporate_Partnership_Director_4__c = atmcpd.get(3).UserId;

                if (atmcpd.size() > 4)
                    ca.Corporate_Partnership_Director_5__c = atmcpd.get(4).UserId;
            }

            if (atmms != null && atmms.size() > 0) {
                if (atmms.size() > 0)
                    ca.Marketing_Solutions_1__c = atmms.get(0).UserId;

                if (atmms.size() > 1)
                    ca.Marketing_Solutions_2__c = atmms.get(1).UserId;

                if (atmms.size() > 2)
                    ca.Marketing_Solutions_3__c = atmms.get(2).UserId;

            }


            if (atmmds != null && atmmds.size() > 0) {
                if (atmmds.size() > 0)
                    ca.Media_Services_1__c = atmmds.get(0).UserId;

                if (atmmds.size() > 1)
                    ca.Media_Services_2__c = atmmds.get(1).UserId;

                if (atmmds.size() > 2)
                    ca.Media_Services_3__c = atmmds.get(2).UserId;

            }
            if (atmdm != null && atmdm.size() > 0) {
                if (atmdm.size() > 0)
                    ca.Digital_Manager_1__c = atmdm.get(0).UserId;

                if (atmdm.size() > 1)
                    ca.Digital_Manager_2__c = atmdm.get(1).UserId;

                if (atmdm.size() > 2)
                    ca.Digital_Manager_3__c = atmdm.get(2).UserId;

                if (atmdm.size() > 3)
                    ca.Digital_Manager_4__c = atmdm.get(3).UserId;

                if (atmdm.size() > 4)
                    ca.Digital_Manager_5__c = atmdm.get(4).UserId;
            }
        }
        update calist;

    }
    global void finish(Database.BatchableContext BC) {

    }
}