In general, a VAT registration can be completed within 2 - 3 days after the date of incorporation. This is on the basis that all the required information is provided and HMRC do not have cause for further checks and questions.
We will be notified initially upon completion, email the details to you shortly after, and the VAT certificate will posted to the trading address (normally your home or office address) within one week. After five days this can be requested online by registering for a Government Gateway ID.
Generally speaking any application that looks as though the business is not operated from the UK or potentially is not a straight forward trade is paused until further information is requested.
If HMRC requests further information then it can take from 14 days for each request to be processed, normally in the format of a questionnaire, sent to the trading address.