What Would You Do?

I would say that professional responsibility means we do this whether we believe or not, however, I don't actually see anything wrong in swapping the task if this doesn't inconvenience anyone else and the resident is happy about it. I have in my time had many conflicts between my own values and those whom I am supporting, but at the end of the day I chose to do a job that involves supporting people, I may not always have the same views or beliefs but I do have a professional responsibility to provide support, if there was a real conflict I would negotiate and find a way of supporting that might not involve me directly, but would give the individual the support they need

Personal belief should not interfere with professional responsibility. Also knowing where to draw the line in carrying out one's professional responsibility is also key.As far as the resident in this case has not asked me to get involved in his/her personal belief, just to sit and ensure they are safe while indulging in their religion belief is as good as it gets for me.

People we support are protected by the Equality Act and they have the right to practice their religion.

I think in this situation there was a sensible approach, quiet and discreet, however if no other staff member was available, the support worker should provide the requested support, without expressing their own beliefs. There should never be any obligation to join in with prayer, but it is their responsibility to provide person centred support.

I would encourage a support worker who had any strong concerns about this part of their role to talk to their manager – other arrangements may be able to be put into place to ensure they are not put into a difficult situation, or maybe support or training can be arranged around understanding cultural differences?

I think there are some excellent points already noted in answers. The persons experience is vital at the heart of this and so ‘staff matching’ feels like a sensitive solution ensuring their spiritual needs are being fulfilled in a positive way.
However it also feels like only part of the answer. As reflexive carers committed to equality, celebrating diversity and assuring inclusion, this is an amazing opportunity to look at our diverse beliefs, seek to understand the other and welcome the chance to support a person in their identity.

It’s understandable to have personal beliefs, but professional care means ensuring a resident’s spiritual needs are respectfully supported , whether by you or a colleague , and those decisions should be transparent and reflective, not silent. Personal beliefs matter, but they shouldn’t result in unmet needs ,the key is ensuring seamless, respectful support.
Boundaries are valid, but they must never limit person-centred care.True professionalism is supporting dignity even when beliefs differ.

My very quick thoughts on this, are that it is akin to the situation when abortion was legalised and some doctors had objections to it. I feel sure that the guidance amounted to '... as a doctor you do not need to provide abortions, but you must point your patient to another doctor who will'.

So, I think

They quietly swap the task with a colleague, but no one discusses it.

and

I think in this situation there was a sensible approach, quiet and discreet, however if no other staff member was available, the support worker should provide the requested support, without expressing their own beliefs

seem [together] to be what I'll describe as 'pragmatically correct'. And, perhaps, in this type of situation 'pragmatically correct' is the best you can do?