If US company with a branch office in the UK what is VAT responsibilities for goods arriving in the UK?

In addition, what are our VAT responsibilities for these goods to be resold in the UK and throughout Europe.

The UK branch will have to pay VAT and any import duty (assuming goods are not VAT and duty exempt, subject to IPR suspension or British return goods) at the time of importation, VAT can then be reclaimed on their VAT return.

Once all duties have been paid and the goods are customs cleared they are in 'free circulation' allowing movement to any other EU member states.

... Adding to Jim P's answer - VAT can be 'zero rated' on an invoice if sold to another VAT registered company i.e. not an individual.

If you import goods into the UK you pay import duties. If you sell those goods on to another company you must charge VAT on the sale of the goods. Depending on your volume of sales you may have to register for VAT.

The UK revenue has a great website www.hmrc.gov.uk which has great free content. You can also call the help desk and get very good advice very quickly.