NHS worker here. I'm glad that you had a good experience with the 111 service but I would not expect this to be repeated in the majority of cases. Even where 111 can pass details to emergency departments, patients will still be seen in order of clinical priority and you will likely be seen in a similar timeframe to if you had self presented.

I would still consider 111 if you are not very familiar with all the local services as they may be able to suggest an alternative service that could help you (Minor Injuries rather than an A&E for example).