These can be tricky without actually being In front of the boiler but it's most likely a diverter valve issue. With the actuator removed, does that operate for ch and hw?
If it does it will be the diverter valve if it dont it will be the actuator in most cases unless the actuator isn't getting...
Put heating only on. Is the lever loose ( in heating position). Turn heating off and water on does it spring back to water position? If the answers are yes does the boiler fire up in hw position with hw only on ? If it does a valve could be shut on cylinder return, if it dont could be...
I've given you the advice your looking for so you have a choice. You can either play with it yourself (which you probably enjoy) or get someone in to fix it!!
In hot water position and with hot water only on, there will be resistance to move that lever to a mid position and if not locked in will spring back. Other than that the valve could just be jammed in heating position.
The little arm as you put it dont move in normal operation. It's there to manually set the valve in a mid position. Does the boiler fire up when you turn the water on? Is you cylinder stat set to 60° ? Have you had any plumbing work done recently? By that I mean some people don't know how to...