VAT on cross-border services is dependent upon the deemed “place of supply” – see HMRC’s notice 741A https://www.gov.uk/guidance/vat-place-of-supply-of-services-notice-741a#sec7 . Any liability to account for VAT arises in that place.
The “general rule” for B2B supplies is that they take place where the customer belongs. However, there is an exception for “land related” services (section 7 of HMRC’s notice), where the place of supply is deemed to be where the land is.
Storage is regarded as “land related” only if the customer is allocated a specific area for their exclusive use. If there’s no specific area allocated for the customer’s exclusive use, the “general rule” applies instead.
You’ll know best whether the storage you’re supplying would meet the criteria for counting as a “land related” service.
– If it doesn’t count as “land related”, you’ll need to charge UK VAT as that is where your customer belongs.
– If it does count as “land related”, you won’t need to charge UK VAT. However you may have some tax obligations in Tanzania!