I would say that in many ways leaf mold and compost are similar in that they are both great soil amendments that add excellent soil structure. However the big difference is that leaf mold only does that, whereas compost also adds nutrition to the soil as well, limiting the need for fertilizers that would be needed if you are only adding leaf mold to your beds.
So if you want to speed up your compost pile add a lot more greens, whether from your kitchen, lawn clippings from you and your neighbours, used coffee ground from your kitchen, friends or coffee shops and although not free, alfalfa pellets that have been soaked add a great nitrogen source along with other micro nutrients and also triacontanol, a naturally occurring growth promoter. Lastly, you need to add a good source of bacteria to speed up the process and build more heat, whether it's finished compost or even a few shovels full of soil.
In the future if you want to get even more into it you can greatly multiply the benefits of compost by using the finished product to make aerated compost tea, which is greatly beneficial to your gardens as a soil drench and to your plants directly as a foliar spray. I generally have a couple of pails going during the summer. I started off using regular aquarium aeraters like in the picture, but I have now upgraded to commercial grade air pumps that put out 45l/min that will support much higher bacterial growth in the tea. Along with that growth also comes the need for more carbs to feed the microbes, so I increase the amount of unsulphured (blackstrap) molasses I use. There is tons of info out there, here's one recipe for a start.
https://fromhungertohope.com/what-kind-of-molasses-do-you-use-for-compost-tea/
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