Yes, Jo, you can part register through a FY - indeed the registration date will be advised by HMRC VAT once they have processed your application.
From that date onwards, you change your invoices to show the issued VAT number on your paperwork, and change the appropriate rate to customers. I think you will find that they always start you from the 1st of a month.
When it comes to working our your turnover and your purchased goods and services, from your VAT date onwards, you exclude any VAT amounts in your turnover or costs so, depending what sort of system to use for management accounting, you may need to make a few adjustments. Your accountant can advise, if you use one.
Are you aware of the Fixed Rate Scheme (FRS) within VAT? Using this scheme, you do not need to account for every little element of VAT on your purchased goods and services, which you net off your invoiced VAT amounts to calculate the VAT amount that you owe the VATman quarterly. Using FRS, you use a flat percentage of your invoiced VAT amount to pay the VATman - let's say 16%. So you charge your customers 20% on your invoices but pay the VATman 16% of that sum. The difference is the notional allowance for your purchased g&s, on which you have paid VAT. The FRS % figure varies according to what type of business you run. It might help make things simpler.