No, it isn't bounded by the access time of the last level in the hierarchy. You could miss in the TLB before you even get to the storage hierarchy. Your TLB miss may even require a page table walk with more misses. You could even get rescheduled because your quantum was exceeded.
You probably think this never happened but it used to always happen. The Linux scheduler used to traverse a linked list of pages (because Linus didn't know any better). Once the number of processes being scheduled exceeded something like 64, it slowed to a crawl consuming most of the quanta just rescheduling.
Yeah, there are finer points to consider to get at an actual bound, and there might be some quirks and infelicities on some platforms. But this is not about competing processes on the system. For matters of asymptotic Analysis the access time is still O(1), i.e. bounded by a constant. If it wasn't, what would it be?
You probably think this never happened but it used to always happen. The Linux scheduler used to traverse a linked list of pages (because Linus didn't know any better). Once the number of processes being scheduled exceeded something like 64, it slowed to a crawl consuming most of the quanta just rescheduling.