Note that FRIDAY merely supplements the suit's abilities, much like a digital assistant. FRIDAY is the equivalent of Siri on an iPhone: Siri is able to set your alarm, but that doesn't mean that you can't set your alarm yourself. Similarly, FRIDAY helps Tony, but that doesn't mean Tony can't do it himself if FRIDAY's unavailable.
The actual operational logic needed to run the suit (servomotors, rocket boots, hand cannons, ...) is housed locally in the suit itself.
If you recall JARVIS and FRIDAY's input, it is often observing more than acually executing. They provide Tony with alerts (suit damage, incoming enemies, things happening in another location, ...)
FRIDAY does not control the suit unless Tony gives an explicit command. FRIDAY merely adds the ability for Tony to give voice commands that are handled intelligently, instead of having to do everything himself. This is useful in cases where Tony has his hands full.
But Tony has executive control over the suit and is able to use the suit to its fullest extent - with the minor caveat that Tony is limited to his own skills and cannot rely on FRIDAY's augmented help.
but his nano suit continues to perform complex tasks with ease
Can you name an example of something that can't be done by Tony himself? Why do you attribute the "complex tasks" to the suit and not Tony himself?
Is there any backup for the AI in the newest suit?
The first real issue is the size of FRIDAY. Not so much the memory space, but rather the massive server park that is hidden from sight. An entire server park cannot be housed in a suit. There are already issue with how much is packed in Tony's suit without considering it also housing the AI itself.
But even ignoring the server park issue, it makes no sense to have a FRIDAY-equivalent AI in the suit. That would mean having two FRIDAYS (one in the cloud, on locally), which means that they are different entities. Part of JARVIS/FRIDAY's allure is that they are a single entity with access to all of Stark's resources.
There is also little benefit to having a local AI. FRIDAY's main goal is to alert Tony of external events, which you can't do when you're offline. Any information an offline FRIDAY is able to access, Tony can access himself as well.
Also, keep in mind how many Iron suits Tony currently has. Are you expecting all of them to each have their own local AI? The synchronization issues between the suits alone are reason enough to simply not house AI locally in the suits.
I remember from the first Avengers film, when Stark guides the nuke through the wormhole, his suit shuts down (lack of cellular communication possibly?).
Fast forward to Infinity War and a similar situation happens when Stark stows away on Ebony Maw’s ship. A call to Pepper cuts out due to range limits.
Between movies, Tony continuously improves the suits. As is shown in the beginning of Infinity War, when Bruce Banner cannot get over the fact that Tony now has nano suits.
Compare this to Iron Man 2, where Tony needed to travel with a rather hefty suitcase which needed to deploy itself to then become Tony's suit. Or the first Iron Man movie, where equipping/unequipping the suit was a cumbersome process.
If anything, the connectivity issue Tony encountered in the first Avengers movie (almost killing him) can be the driving force behind Tony improving the suit so this does not happen again.