perhaps the attached datasheet of a MIL-STD-1553B remote terminal sheds some light on the required properties: It is clocked with 16 or 24 MHz, so I would look for microcontrollers with risc architecture, capable of a clock frequency of at least 20 MHz. For example, both in AVR and in PIC families (both by Microchip now) are a lot of members working at 20 MHz or more. I myself would start with an ATtiny861A, being prepared to switch later to an ATxmega if necessary or to add a second microcontroller, the first dedicated to the low level task of data decoding with some fault tolerance, and the second working on a higher level, e. g. differentiate between command words, data words, and status words. This second microcontroller might even be slower than the first.
Depending on the kind and quality of your signals, you could drive the microcontroller pins directly, or use transformers as in Fig. 1 of the attached document.
How much computing power you need, depends, of course, also on the intended further processing of the sampled data: Is it enough to store some kByte in the internal RAM, and tranfer the data later on (not in real-time) to a computer? Or is your goal a stand-alone system which can store a large amount of data, e. g. on a SSD? In the second case, the combination of a microcontroller for data decoding and something like the Raspberry Pi running an off-the-shelf OS plus a simple application program might save a lot of programming work.
If you're set on an MCU, enable capture/compare interrupts on both rising and falling edges, and calculate the bit-data based on the elapsed time between interrupts and the type of transition event.