Disclaimer: This answer is largely based on personal experience and anecdotes and written from the perspective of a Simplified Chinese user.
For English: wpm/cpm, including punctuation. Conversion: 1 wpm = 5 cpm.
For Chinese: cpm, including punctuation.
It all depends on the text you're typing, the device you're using and the particular method of input you're using. To understand typing speeds on mobile, you first have to understand...
Casual/Professional typing with physical US keyboard
Apart from the amount of training, the speed difference among various English typists come from the keyboard layout. There's no "IME" so to speak, because whatever character you type will end up on your text-based editor. Two common layouts are QWERTY and Dvorak simplified.
Usually with QWERTY keyboard but with specialized input methods (IME), including but not limited to 拼音、双拼、郑码、二笔、五笔 that turn a sequence of keyboard inputs into one or more Chinese characters or symbols. To make typing faster, modern IME often have native, on-the-fly or "cloud" dictionaries where common phrases are given an input sequence.
As I'm only familiar with 拼音 and 五笔, I'll use them in the following comparisons:
For text that are archaic or severely lacking in multiple-character phrases, 拼音 usually is the slowest due to the sheer amount of characters you have to choose for each pinyin—a phenomenon called 重码. When using 五笔 to input characters one by one, the frequency of 重码, a.k.a. 重码率 is fairly low. The effort to learn 拼音 without memorizing the selection in 重码 is way lower than learning 五笔 without memorizing the selection in 重码, but 拼音 users typically max out at 60 cpm, while 五笔 users average at 60 cpm.
For text that are rich in words and phrases, modern IME can make 拼音 users type at least faster than inputting single characters. On the other hand, 五笔 users relying only on non-updating built-in dictionaries also get a speed boost. Both types of users can reach 120 cpm1.
Casual typing with touch-screen keyboard
This section could partly explain mobile phones with a physical keyboard.
This is where the world slows down. To fully utilize the screen, one would think about using the 26-key layout on smart phones, but because the smart phones are small and one is only tempted to use a maximum of 2 fingers (as opposed to a large keyboard and a maximum of 10 fingers in use) mis-clicks occur more frequently. To cope with this, layouts that lump multiple characters into one key are designed. As long as you know one of the keyboard IMEs, one would type faster than writing using character recognition IME. Modern mobile IME use dictionary and prediction technology as well, but well-trained physical keyboard users who haven't trained in touch-screen keyboards will experience a significant speed drop.
Professional typing with specialized keyboard
A.k.a. steno-typing. These are not used in everyday scenarios so I'll make it short.
World record for Steno keyboard stands at 375 wpm. Steno-typing typically works by parsing a sequence of simultaneous multi-key inputs into one or more words.
Similarly, stenotyping systems such as 亚伟速录 exists, and a user using its specialized keyboard maxes out at 500 cpm.
1 Anecdotally, I know a 五笔 user who averages at 180 cpm on text that contain 1/6 archaic Chinese.