You are legally obliged to record movements on and off your holding of live animals.
Apart from that there is no requirement to account for animals born or dying (or being killed!) on your holding.
However there is a requirement to record once a year (on a date which you choose) the maximum number of pigs that have been on your holding in the previous 12 months. Defra cannot say why this rule exists, and it is totally pointless, but they won't take it out of the legislation without being given a good reason to (they tell me that being pointless is not a good reason!) so it will continue, so much for red tape reduction.
So that's all you need.
So if anything is born or dies (or indeed you kill for home consumption) then this is not a legal requirement to record this. However you are not permitted to bury animals or parts thereof on your holding.
If animals dies and you use fallen stock, then they will give you some paperwork, which in the event that an inspector who does not know the law asks, you could wave at them, but this is not needed.