Maybe.
In workplaces where it is difficult to guarantee social distancing, employers could introduce a requirement that employees who have a device provided by the employer download a contact-tracing app. The justification being that it is required to protect health and safety and minimise sickness absence.
If the employer intends to process personal data produced as a result of the employee using the app, it will need to have a lawful basis for processing this data and comply with the GDPR and the Data Protection Act 2018. This would include conducting a prior assessment of the risk of the processing. It is difficult to rely on consent in an employment context as the basis for processing data due to the inherent imbalance in the relationship between the employer and employee, and any legitimate interests grounds for such processing may be harder to justify where the nature of the solution proposed is not proportionate.
The issue is not straightforward. If employers are going to insist that employees use an app, they need to be confident that the app is secure so that health data will not be used for purposes other than as intended. It is questionable whether employers can require employees to use the app outside work, which presumably is essential for it to work. Doing so could infringe employees' privacy.
In practice, the employer will be forced to rely on the employee self-declaring if they have come into contact with someone who is infected. If employees who are required to self-isolate are paid statutory sick pay (SSP) this could be a deterrent to an employee who should self-isolate but has no obvious symptoms.
The ICO and government are likely to publish further guidance on contact-tracing apps and this will be helpful for employers considering the introduction of such a requirement to its workforce, given the complexity of the issue and the factors that need to be balanced.