This is a tricky one, as proper names often don’t follow the general spelling/pronunciation rules of the language - think of Cholmondley pronounced 'chumley', for instance! And there are many variations of the spelling of Shakespeare's name. According to David Kathman, who collated them all, we have the following:
Non-literary references (1564-1616)
Shakespeare 71
Shakespere 27
Shakespear 16
Shakspeare 13
Shackspeare 12
Shakspere 8
Shackespeare 7
Shackspere 6
Shackespere 5
Shaxspere 3
Shexpere 2
Shakspe~ 2
Shaxpere 1
Shagspere 1
Shaksper 1
Shaxpeare 1
Shaxper 1
Shake-speare 1
Shakespe 1
Shakp 1
Literary references (1593-1616
Shakespeare 119
Shake-speare 21
Shakspeare 10
Shaxberd 4
Shakespere 4
Shakespear 3
Shak-speare 2
Shakspear 2
Shakspere 1
Shaksper 1
Schaksp. 1
Shakespheare 1
Shakespe 1
Shakspe 1
For the first syllable, there are clearly two types, with an -e and without an -e, and this is an important difference, as the presence or absence of an -e was one of the signals of the contrast between a long and a short preceding vowel (you can read more on the spelling background in my new book Spell It Out).
The Shakespeare spelling is overwhelmingly the predominant one. Shake rhymes with make, take, and quake in the canon, which clearly suggests a long vowel, and this would in original pronunciation be a mid-open front vowel, approximating to a long version of the modern vowel we hear in RP pet. On the other hand, the Shak, Shack, Shax series clearly suggests a short front vowel, as in RP back today. How to reconcile the difference? There are many spelling variations which suggest that the OP short vowel of back was higher at the front of the mouth than it is in RP today, closer to the short /e/ of bet: we see, for example, acts written as ectes, and there are several other instances. There are also many rhymes which show that the short /a/ vowel must have been close to short /e/, such as back rhyming with neck in Venus and Adonis.
If we start with Shake, this would have had a long /e/ vowel, but - as with all long vowels - it would sometimes be pronounced rapidly, and be heard as a short vowel, and spelled accordingly. If we start with Shak, this would have had a short /e/ vowel, but - as with all short vowels - it would sometimes be pronounced slowly, and be heard as a long vowel, and spelled accordingly. Either way, we end up with the same result - a vowel sound which is roughly what we hear in share in RP. (Phonetic symbols don't always come across easily in blogs, but the relevant symbol for this vowel is the mid-open front one - /ɛ/) There's also the option that a Warwickshire regional pronunciation would have affected the length, but there's no firm evidence about that.
For the second syllable, the main point to note is that the /r/ would have been pronounced at the end. All sources agree on that. As for the vowel, the spellings suggest a long vowel, as in spear. But when we look at spear (and similar words) we find it could rhyme with there (in Lucrece and Venus, for instance) and similar-sounding words, and it this which doubtless motivated such spellings as -pere, -berd, and so on in the name. The vowel may also have had a shortened and centralised form, being in an unstressed syllable. So it would have been roughly what we would hear today in (long) spare or (short) spur.
In short: I would say the evidence points to something like /shɛ:kspɛ:r/, with /shɛksper/ or /shɛkspur/ as more rapidly said alternatives.
