Yes, 20 years ago, when I was a Reception class teacher, a child soiling themselves was a rare occurrence. Perhaps once or twice a term. And usually due to an upset stomach. I am not including children with additional needs in this.
If it did happen and a child didn't have an LSA, our admin team would call the parents and ask them if they would prefer to come in/send someone. We would explain that most children would prefer to be changed by a family member (which is true).
Almost all children would be changed by a family member. Promptly as most families lived very close to the school and had extended family around. If that couldn't happen, then we would change. Always at least 2 fully qualified, permanent members of staff. Not only the TAs.
In my 5+ years in Reception, I probably changed a soiled child once a year. Wee accidents we could simply help the child access clean clothing, wipes etc.
The difference is that it was fully expected that children were toilet trained before starting Reception unless there was an identified reason or investigations ongoing.