I was going through some online articles and I'd like to thank @Josh61 for the right references. I found this detailed write-up on Word Origins from OED by Richard Holden. (I think I now know the reason why top EL&U users strictly stick to OED definitions)
The article: http://public.oed.com/aspects-of-english/word-stories/digital/
What distinguishes digital from many other terms associated with high technology is that it’s not a new word. In the newly revised OED entry, the earliest evidence—in the sense ‘designating a whole number less than ten’—dates from the fifteenth century. OED‘s original entry, published in 1897, does not record this sense. Instead, it covers senses corresponding to another sense of digit, such as: ‘of or pertaining to a finger, or to the fingers or digits’—evidence for which goes back to the seventeenth century. But for most of its history, digital was a relatively unimportant term: it wasn’t until the early to mid-twentieth century that the word became more significant and widespread.
In the late 1930s and early 1940s, the work of mathematicians and engineers led to the development of a new type of computing machine. As opposed to earlier analogue devices, which used a continuous quantity (such as voltage) to compute the desired quantity by analogy, these new machines operated upon data that was represented as a series of discrete digits. For example, in such a system the letter A might be represented as the binary sequence ‘01000001’ (as it is in the ASCII encoding scheme).
Being composed of such sequences of digits, such data (and so any machine making use of it) was hence said to be "digital". Digital computers were generally considered more adaptable and powerful than their analogue counterparts, and digital computing became dominant: the computer you are reading this article on will certainly be a digital one, as will probably any other computer you have ever used. The sense of digital relating to this was covered in OED2 (1989) by the definition, ‘of, pertaining to, or using digits; spec. applied to a computer which operates on data in the form of digital or similar discrete elements.’
Now, to answer your questions...
1. Was Alan Turing aware of the term digital computer? Did he actually ever use it himself?
Yes. Alan Turing developed the Turing test, and uses the term often in his seminal paper "Computer Machinery and Intelligence".
"Are there imaginable digital computers which would do well in the imitation game?
2. Could one describe the British Bombe as being a digital computer? Wikipedia calls it an “electromechanical device”
I would say No. At least not a "digital computer". It's called electromechanical because it operated on electricity and had rotors, wires and plugboards that carried out the deciphering work. Importantly it did not operate on 0s and 1s, which is the definition of a "digital computer".
(noun). an electronic computer in which the input is discrete rather than continuous, consisting of combinations of numbers, letters, and other characters written in an appropriate programming language and represented internally in binary notation
The ENIAC is widely believed to be the first ever "digital" computing device.
It was Turing-complete, digital, and could solve "a large class of numerical problems" through reprogramming
You could probably say it was an "analog computer"!
(noun) a mechanical, electrical, or electronic computer that performs arithmetical operations by using some variable physical quantity, such as mechanical movement or voltage, to represent numbers
3. When was the term digital computer actually coined?
The exact time and person who coined it remains a mystery. Etymonline suggests that both "digital" and "computer" originated in the middle of 16th century but the combined term, i.e., "a digital computer" might have its origins at the beginning of the 20th century, and a spike in usage by late 1940s, according to Ngrams.