I don't understand your use-case for this, but you'll need to get your other SOQL query outside the for-loop. Also, I'm assuming (maybe incorrectly) that your inner SOQL should be limited to the campaign member records you initially selected. I don't understand why you'd want to query all campaigns for all time to get the contact count. Frankly, I don't understand why you'd initially want to query all campaigns for all time for a specific campaign type either.

It sounds like you want each campaign member to indicate the total number of contacts in the campaign that have the same account as the contact on the campaign member record. To store a value on an SObject record, even if you're not saving it to the CampaignMember object, you'll need that field to exist on the SObject. So *if* you create a "contactCount" field on the CampaignMember object, then you code code it like this:
 

@AuraEnabled
	public static List<Object> getContactCMCount() {
		List<SObject>  result = new List<SObject>();
		result = [SELECT
					Id,
                    contactCount, // you need the field to be retrieved in the query to access it
					Contact.Id,
					Contact.Name,
					Contact.AccountId,
					Contact.Account.Name 
				FROM CampaignMember
				WHERE Campaign.Type = 'Programme Application'
			];

        // Create a map of Account Id & Count of contacts & initialize to 0
        Map<Id, Decimal> contactCountMap = new Map<Id, Decimal>();
        for(CampaignMember cm : result) {
           contactCountMap.put(cm.Contact.AccountId, 0);
        }

        // Query contact members and group by contact's accountid to update the map
        for(AggregateResult ar : [
            SELECT Contact.AccountId accountid, count(Id) nbrcontacts
            FROM CampaignMember
            WHERE Campaign.Type = 'Programme Application' // or not, if you want all campaigns
              AND Contact.AccountId in :contactCountMap.keySet()
        ]) {
           contactCountMap.put((Id) ar.get('accountid'), (Decimal) ar.get('nbrcontacts'));
        }

        // Assign contactCount to each campaign member		
		for(SObject currentCM : result){
            Contact currentContact = currentCM.getSObject('Contact'); 
            currentCM.put('contactCount', contactCountMap.get(currentContact.AccountId));
		}

		return result;
	}

If you don't want the new field in the CampaignMember object, then you'll need to create a wrapper object that has the CampaignMember record as one member (or each element of CampaignMember you need) and the count as another.  For example (I've renamed the initial SOQL results because it's no longer the result that is returned):

@AuraEnabled
	public static List<Object> getContactCMCount() {
		List<SObject>  campaignmbrs = new List<SObject>();
		campaignmbrs = [SELECT
					Id,
                    contactCount, // you need the field to be retrieved in the query to access it
					Contact.Id,
					Contact.Name,
					Contact.AccountId,
					Contact.Account.Name 
				FROM CampaignMember
				WHERE Campaign.Type = 'Programme Application'
			];

        // Create a map of Account Id & Count of contacts & initialize to 0
        Map<Id, Decimal> resultMap = new Map<Id, Decimal>();
        for(CampaignMember cm : campaignmbrs) {
           resultMap.put(cm.Contact.AccountId, new CampaignMemberWrapper(cm, 0) );
        }

        // Query contact members and group by contact's accountid to update the map
        for(AggregateResult ar : [
            SELECT Contact.AccountId accountid, count(Id) nbrcontacts
            FROM CampaignMember
            WHERE Campaign.Type = 'Programme Application' // or not, if you want all campaigns
              AND Contact.AccountId in :contactCountMap.keySet()
        ]) {
           CampaignMemberWrapper wrap = resultMap.get((Id) ar.get('accountId'));
           wrap.contactCount = (Decimal) ar.get('nbrcontacts');
        }

		return resultMap.values();
	}

// wrapper class to accomodate a count of contacts with the same account as the campaign member
public class CampaignMemberWrapper {
    CampaignMember campaignMbr {get;set;}
    Decimal contactCount {get;set;}
    public CampaignMemberWrapper(CampaignMember cm, Decimal cnt) {
        campaignMbr = cm;
        contactCount = cnt;
    }
}