What CPU does the F-16 use?
MIL-STD-1760A is simply the standard that a processor must conform to, and is not an actual model of CPU.
There are actually a lot of independent computers in a military aircraft, each with specific responsibilities. They are usually linked up via networks, typically over MIL-STD-1553 busses for older platforms. The radar has its own processor(s) and architecture(s), as does the flight control module, the fire control module, the navigation system, the munition controllers (sometimes one per weapon station/pylon), the display controller, the mission computer, et cetera. All of these come from different OEMs, and therefore use a variety of hardware. The different subsystems present and their exact functionalities depend on the design and requirements. A subsystem may itself be composed of multiple computers within their own network, independent of the main aircraft networks.
The F-16 uses Raytheon's Modular Mission Computer (MMC) for its mission computer, with the following specs:

The first generation, the MMC3000 relied on MIPS R3500, a 32-bit chip processors with 110,000 transistors and running at 12 MHz. We found this chip on the end of the 1990s HP9000 computers. The computer had 4 MB of memory RAM.
The MMC5000, the second generation of the F-16 computer, still relied on a chip MIPS, a RM5260 who was noticeably faster (with a 133 MHz to 150 MHz clock frequency) and above all it is a 64-bit chip. The memory was more than doubled, with 10 MB available.
The MMC7000 that equips the more modern F-16, including all F-16 Europeans who benefited from the MLU (mid-life Update) modernisation, has always that 10 MB of memory but its RM7000A processor, designed in the early 2000s, works between 300 and 400 MHz.
Source: http://www.4erevolution.com/en/ordinateur-de-bord-f-16/

I highlighted the mission computer because I think of the mission computer/controller/processor (exact terminology depends on the platform) as the brains of a military aircraft. They tend to be the bus controller of the main aircraft network, aggregating inputs from other subsystems and computing/providing information to the pilot to help them carry out their mission.

To the second part of your question, I assume you mean faster or more powerful CPUs when you talk about modern CPUs. Modern here depends on context. CPUs used In aircraft will never need the type of clock speeds or memory capacity you see in consumer computers, for a few reasons. Firstly, there is actually a lot of processing capacity distributed over the various subsystems - the need for a single powerful chip only arises when you demand that a single chip do everything. Secondly, when the requirements of subsystems are as specific as they as in military platforms, the software can be very compact and efficient, unlike consumer software (which require a lot more code for things like architecture adaptors, multi-tasking logic, and thousands of device drivers) - you trade off unnecessary flexibility in return for reliability and efficiency.
The multi-functional displays of military aircraft (and also commercial ones) focus on conveying the necessary information in a clear and concise manner. The rendering requirements are actually quite primitive compared to modern 3D games - it can get by with less processing power than a computer from the 1990s.

EDIT: Elaborated answer, following user3528438's comment.