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Obviously an employer absolutely can't require employees to vote for any particular candidate (or even to vote for any candidate rather than spoiling their ballot), but if it was included as a clause in employees' contracts that they must (in company time) attend the appropriate polling station and "participate" in some defined set of elections (eg "all local and national governmental elections and referenda"), and that failing to do so was a disciplinary offence, would such a clause be enforceable? Would an employer legitimately be able to fire someone for failing to comply?

Looking for answers in relation to the UK.

Enforcement by firing a person could be a problem. There are specific allowed reasons to fairly fire an employee, which does not include "failure to vote". The description of unfair reasons includes, as an example, joining a trade union, and other actions that have some imaginable connection to the workplace. But the government has not clearly declared that political expression (or its lack) is or is not fair grounds for dismissal. Such a firing would be subject to scrutiny under the unfair dismissal doctrine, which means that the two sides would seek supporting analogs in UK case law.

The factors that favor a "fairness" finding are heavily weighted towards the legitimate business interests of an employer. This article analyzes free speech rights in connection with the unfair dismissal doctrine. As an example, in Smith v. Trafford Housing Trust, the claimant was punished (demoted) -- unfairly, the court fount -- for expressing a political viewpoint on Facebook. The court noted that the outcome would have been different if claimant had promulgated his views in the workplace. If an employees action brings a business into disrepute, perhaps a dismissal could be found to be fair. But failure or refusal to vote does not have that effect: it is not a legitimate business interest of the company, whether the employee votes.

Good answer already from user6726, but an additional consideration why the original contract could be ruled unlawful would be if an employee's belief in a right not to vote could be considered a "philosophical belief" under the Equality Act 2010. According to ACAS, criteria for this had been defined at an earlier (2009) tribunal.

The ACAS link compares religions with other philosophies, but there are also religions who advocate political disengagement. If an employee was politically disengaged for religious reasons, it would be easy to argue that the contract constituted discrimination that was prohibited by the Equality Act.