We present LL-MEC, the first open source Low-Latency Multi-access Edge Computing (MEC) platform enabling mobile network monitoring, control, and programmability while retaining compatibility with 3GPP and ETSI specifications. LL-MEC achieves coordinated resource programmability in end-to-end slicing scenarios by leveraging SDN towards an appropriate allocation of resources, thus drastically improving the performance of slices. We evaluate LL-MEC in three practical case studies, namely, (i) end-to-end mobile network slicing, (ii) RAN-aware video content optimization and (iii) IoT gateway, and show that it achieves a 2-4x lower user plane latency compared to LTE, while enabling low latency edge applications to operate on a per millisecond basis. Also, we highlight the benefits of RAN-aware applications in improving user Quality of Experience (QoE), showing a significant user latency reduction along with a much lower variability compared to legacy LTE. Last, a compatibility evaluation of LL-MEC over the OpenAirInterface real-time LTE platform demonstrates the scalability merits of LL-MEC due to the use of an OpenFlow Virtual Switch for the user plane function, rather than a Linux kernel in typical LTE setups.