Out of a group of 500 people over the age of 80 one dies within a week.

→ This means it will happen often that a person who was just vaccinated dies within a week.
Not due to the vaccine, but due to the high death rate of the people who are vaccinated first.

data in next tweet

The annual death rate for those over 80 is 10.75% in England & Wales

This means the chance of dying in any week is 0.206%.

[= 10.75% per year / 52 weeks]

So that out of a group of 485 people over 80 we have to expect one person to die within a week.
[100/0.206=485.4]

We will carefully monitor whether the various new vaccines have serious side effects.

But it's important to keep the high death rate of elderly people in mind as some might be tempted to conclude that a death on the next day was due to the vaccine even when it was not.

tl;dr: Correlation does not imply causation.