Before you get started on what rules apply to your product you need to know how that product is classified in the EU. See https://www.gov.uk/classifying-tobacco. In essence if an electric cigarette is imported with a packet of cartridges, the whole set is to be classified together to commodity code 8543 70 90 99 as an electrical product not elsewhere specified. However, a nicotine cartridge containing a preparation of nicotine intended to assist smokers to stop smoking is classified to heading 3824 90 97.
Much depends on the make up of the cartridges e.g. how much if any nicotine they contain and what other chemicals they contain.
The normal system in the EU is that if the exporter is tax registered in the country of export and the importer is tax registered in the country of import then VAT should not be charged by the exporter. The importer would then self-report and assess his tax liability and depending on his business the VAT due may well be offset - in effect becoming non payable.
If the exporter is tax registered but the importer is not tax registered in the country of import then the exporter must charge UK VAT to the importer who will not be able to reclaim it.
If you (A UK company) are buying product from a UK tax registered company in the UK this is a local or domestic transaction and is subject to VAT. This is a separate transaction to your eventual sale to a customer somewhere else in the EU.
If you are exporting and your customer is importing you would not need to become involved in the VAT of the importing country. You need to ensure your choice of INCOterm reflects this.
A company registered in Gibraltar is not going to work well as the exporter from the UK - for it to be involved you would really need to be exporting from Gibraltar.
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