I have said before on AB that even if every person in the UK, including all the members of the Commons and the Lords, wanted to restore Capital Punishment in the UK, it could not be done.
Not, that is, unless we repeal the 1998 Human Rights Act and withdraw as a signatory from the European Convention on Human Rights (ECHR). Since being a signatory to the ECHR is (more or less) necessary for membership of the EU, it would also mean leaving the EU.
The derogation process that jake mentions applies mainly to new directives. (An example is the “opt out” Britain achieved to the working time directive which allowed hospital doctors top continue working hours in excess of the directive). These are usually temporary arrangements provided for transition periods to allow individual states to make adjustments where there circumstances dictate.
Such derogation facilities are not available for provisions under the ECHR. You are either in or you are out. The ECHR (Protocol 6) requires signatories to restrict the use of the death penalty to times of war or imminent threats of war. I have not seen the film in question but if the makers say the ECHR was “amended” to accommodate an execution it is pure fiction.
The issue of Capital Punishment is fairly straightforward – it is forbidden, and in any case the UK abolished it in 1968 under domestic law. What is not quite so straightforward are some of the other provisions of the ECHR (and the 1998 HRA) which are deliberately vague and open to wide interpretation by the courts. It leads to situations where, for example, “travellers” find they are able to ignore UK planning law (to which the rest are subject) because to force them to comply with it would infringe their Human Rights.