In the first scenario, where the goods are manufactured in the UK and shipped directly to the customer in Northern Ireland, there is no VAT applicable on the sales invoices raised by Company A. This is because Northern Ireland is still part of the EU VAT area, even though it is not part of the UK. Therefore, the goods are subject to UK VAT, which is 0% for goods moving between Northern Ireland and the rest of the EU.
In the second scenario, where the goods are shipped from the UK to a freight hub in Southern Ireland, who then arrange shipment to a customer in Northern Ireland, there is VAT applicable on the sales invoices raised by Company A. This is because the goods are technically imported into Ireland, and then exported to Northern Ireland. The VAT rate on the sales invoices will be 23%, which is the standard VAT rate in Ireland.
You can also use this VAT Calculator tool
that I find very useful during research.