I think nursing care is always a complex interpersonal process, especially wrt to intimate care and everyone brings with them their own personal worries and issues. A good part of nurse training covers consent and equality and diversity, and also treatment and personal care. Everyone is vulnerable, especially if they are worried or ill.
I've been a nurse for the NHS for quite a while, and I've seen it change quite a lot. Usually the most frequent issues I've seen are sexism, rascism and homophobia. There has also been quite a lot of classism.
Sometimes people clash for whatever reason, and whenever possible, whichever nursing team I've been on, if there's a problem we always try to find a solution within the team. It's usually just a personality clash but sometimes it's outright rascism/homophobia/transphobia in which case yes we are careful to take steps to try and look after both the member of staff and work out how best to care for the patient. If a patient expresses that they don't want someone looking after them, we will try to accommodate.
The trouble is, if you are rascist, homophobic, transphobic, sexist or classist, it's difficult so actually say so because it's culturally unacceptable!
Negotiating personal issues is difficult, and from my own personal experiences I know how difficult it is to say - "I'm really not comfortable with this", or "can someone else look after me" because you don't want to hurt the person's feelings, or appear to be a "difficult patient", so I can really sympathise.
Many people don't care / mind who is looking after them as long as the care is competent, professional and compassionate but if you're really frightened about trans* people, then the only thing I can think of is really to be open about it and specify who you would prefer to look after you.