Here are some factors which appear* to be considered.
*Through observation and experience when making/seeing requests to hold connections.
The existence of a guard. If there isn't a guard, it is much more difficult for people to make contact with the operator to hold connections.It is very rare in former Network SouthEast land for connections to be held.
Also, it varies by guard. Some are more proactive than others when enquiring if people are making common connections, for example.
Last connections of the day. It is more common to hold those than others. TOCs would rather slightly delay a train than pay the bill for a hotel/taxi for a long distance. For example, I've often successfully requested that either the 20:15 from Inverness to Glasgow or the connection from that service at either Perth or Stirling to Edinburgh is held (they're the last trains of the night) because of late Far North running. I've yet to had the pleasure of being chucked on the sleeper (which I gather is the last resort if there is a significant delay).
Number of passengers. If there is a significant number of people making the connection, the service is more likely to be held.
Same TOC transfer. TOCs will be more willing to delay their services for their own passengers than they are for another TOC's.
Operational reasons. What effect will the delay have on other services? How big will that effect be?
Length of time until the next connection.
The TOC itself. Some TOCs appear to be much more willing to hold services than others.
It's worth noting that it is a massive lottery. Since guaranteed connections were abolished, there has never been a single we-will-hold-your-connection-if... rule.