Nine, can you be more explicit in discussing the date variable formatting? I believe that's precisely what I'm doing. The actual code looks like this:

// CALLING ROUTINE
// Instantiate the class
AccountTeamBatch ATB = new AccountTeamBatch();

// Capture the last run date
ScheduledApex__c lastRun = [SELECT LastRun__c
                              FROM ScheduledApex__c 
                             WHERE Name = 'ApexRun'];

// Assign the value to a variable in the class
ATB.lastRunDT = lastRun.LastRun__c;

// Create the query string. I've tried it two ways. 

// Approach 1: define a query string using the formatted date. 
// This is what all the documentation says is the right way to 
// query with a datetime string. Unfortunately, it results in 
// the error "no viable alternative at character ' '", pointing
// to the ":" as the culprit.
ATB.query = 'SELECT ID, Account__c, IsDeleted ' + 
            'FROM AccountTeam__c ' +  
            'WHERE Role__c IN ('Exec (MDM)\', \'Infra Exec\', \'CT\') AND ' + 
                  'LastModifiedDate > ' + lastRun.lastRun__c.format('yyyy-MM-dd\'T\'HH:mm:ss') + ' ' +
                  'ALL ROWS';
// query variable ends up looking like this:
// SELECT ID, Account__c, IsDeleted FROM AccountTeam__c WHERE Role__c IN ('Exec (MDM)', 'Infra Exec', 'CT') AND LastModifiedDate > 2010-12-29T21:17:00 ALL ROWS

// Approach 2: same definition, but use variable substitution
// This results in an "Attempt to de-reference a null object" error
ATB.query = 'SELECT ID, Account__c, IsDeleted ' + 
            'FROM AccountTeam__c ' +  
            'WHERE Role__c IN ('Exec (MDM)\', \'Infra Exec\', \'CT\') AND ' + 
                  'LastModifiedDate > :lastRunDT ' + 
                  'ALL ROWS';

// CLASS DEFINITION
global class AccountTeamBatch implements Database.Batchable<sObject>
{
	public datetime lastRunDT;
	public String query;

	global database.querylocator start(Database.BatchableContext BC) 
	{

		// It never seems to hit this debug statement. At least
		// it never shows up in a log, and the query appears not
		// to run.
		system.debug('start: ' + query);
    
		return Database.getQueryLocator(query);
	}
	
	global void execute(Database.BatchableContext BC, List<sObject> scope) 
	{

		// Do some stuff; never gets here as best I can tell
	}

	global void finish(Database.BatchableContext BC) 
	{
	}
}

FYI, the way I'm tracking the last run is simply by creating a custom setting with nothing more than a date field as its content. I then update the date field at the beginning of my run, so it always remembers the last time it ran. Retrieving that date is the first statement after instantiationl, in the above code.