Timing attacks work when code tries to be fast, taking shortcuts where possible. There's a reason we talk big-O instead of big-Theta: we gladly accept when an algorithm finishes early. This is, in this case, highly undesirable. Shortcuts are great, everywhere but crypto; not unlike recursion being great, unless it's embedded.

A language where every expression takes constant-time to compute seems like a solution. No short-circuiting, no variable runtimes. Don't use sorting algorithms like insertion/quicksort where best-case & worst-case are very different; use selection/merge where best-case = worst-case = average-case. Use constant-time hash functions. Caching is a very prevalent (and somewhat invisible) shortcut that needs to be solved (how can a language disable caching?).

Obviously this isn't fast but you can't have both.

Merge sort is still pretty fast. It isn't the fastest. But safe > fast when it comes to reliability and security.