You have a number of considerations here. You should consider what is considered good business practice, what your contracts of employment require you to do and the level of risk you are prepared to take.

The first consideration is your current recruitment policy. If there are contractual obligations in place regarding recruiting for new positions these must be followed. Therefore, if this necessitates advertising, you should do it.

Good business practice would suggest that you advertise, shortlist and interview for every role available within your business. This ensures everyone is given a fair chance as well as ensuring your decision is the best one that you can make in the circumstances and is demonstrably fair. Obviously, as you understand this is a time consuming process however it is considered best practice.

Ultimately, it is up to you if you choose to advertise. Simplistically speaking, you are not required to advertise a role and are free to appoint someone internally in the new role. However, while no legislation exists to demand this is done, there are some risks to not advertising. You could leave yourself open to a discrimination claim from someone both internally and externally, should they find out about the post and the lack of opportunity to apply. You also do not state whether there are other candidates within your business interested in the role? If so, you should consider any application from them to avoid a grievance being aired from this employee(s).