Ask your admin to execute following code snippet. As admins can update records, even when they are locked.
This is for Account, but make changes for your object and replace fieldName with your respective field
Map<id,Account> accounts = new Map<ID,Account>([Select {fieldName}, id From Account]);
Map<ID,Date> approvalTimeStampMap = new Map<ID,Date>();
For(ProcessInstance approval : [Select createdDate,TargetObjectId From ProcessInstance Where TargetObjectId IN:accounts.keySet()){
approvalTimeStampMap.put(approval.TargetObjectId, approval.createdDate);
}
List<Account> accountsSubmittedForApproval = new List<Account>();
for(Account acc : accounts.values()){
acc.{fieldName} = approvalTimeStampMap.get(acc.id); // Do string conversion if your field is a String
//Keep a track of number of records you are processing
accountsSubmittedForApproval.add(acc);
}
//Size should be less than 10k
update accountsSubmittedForApproval;
Hope this helps :)