On the Reading section of the new SAT, you'll have 65 minutes to complete 52 questions. If we take this at face value, it means that you have 1.25 minutes (1 minute 15 seconds) for each question on average. All questions on the Reading section are now passage-based, so you don't have to worry about Sentence Completion questions anymore. However, this can also complicate things because you'll be spending a significant amount of time reading passages before you answer questions.
There are five question groups total in the Reading section (four individual passages and one set of paired passages). This means an average of about 13 minutes per group. Depending on your reading style, you might devote more or less of this time to reading vs. answering questions. If we factor in the reading component, the actual time you spend on each question should be far less than 1.25 minutes. I'd recommend limiting your time to around 45-60 seconds on each Reading question so that you have enough leeway to read and understand the passages.
For Writing, you are expected to answer 44 questions (4 passages with 11 questions each) in 35 minutes. Again, if you take it at face value, this comes out to around 45 seconds per question, on average. Although the Writing questions are also now completely passage-based, you don't necessarily have to spend much time reading the passages before you answer the questions. I'd recommend that you initially allot no more than 30 seconds to each Writing question so that, once again, you give yourself enough time to check over your answers and reread sections of passages that may have been grammatically or stylistically confusing to you.
Reading: 13 minutes per passage group, 45-60 seconds per question
Writing: 8 minutes 45 seconds per passage group, 30 seconds per question
On all sections, if you find yourself spending more than the above amount of time on a question, move on and come back to the question later to ensure that you have time to answer any less challenging questions further along in the section.
For more information on the timing of the new SAT, be sure to read our article that breaks down exactly how long the SAT is (including breaks and the essay).
Sources: College Board, PrepScholar Blog