First the simple bits…
The sequence <quaw> is always pronounced kwɔː, and in fact occurs in a very few words. Squaw and squawk are the only ones I can think of at the moment.
The sequences <ware> and <quare> are pronounced with the vowel eə. Aware and square are examples.
Now things get a bit more complicated…
There are many words which contain one of the sequences where the vowel is ɒ. Here is a list of some of these:
quaff quality quantity quarrel quarry quash squab squabble squalid squash swallow swan swap waft wallow wan want warren wash what watch
The word water is an oddity. It is of course pronounced ˈwɔːtə.
As you can see from the above list, if one of the sequences is followed by <r> or <rr> or <l> or <ll>, which in turn is followed by a vowel, then the vowel in the sequence is ɒ. However, if a consonant or nothing follows then the vowel in the sequence is pronounced ɔː, as in:
quart quarter squall sward swarf swarm walk wall war ward warm warn warp wart wharf
The word qualm is another oddity. It is pronounced kwɑːm.
There are some words where the vowel is æ. Here are the examples:
quack quango swag swank thwack wag whack wax
These all have in common the fact that the next sound is one of k ɡ ŋ. The words swam and wham also have æ. Thanks to Alex Rotatori for reminding me that the wonderful word quagmire has two competing pronunciations: one with ɒ and the other with æ.
Finally, here are some words where the vowel is eɪ.
quail quake quaver wafer wage wake waif waist wail waive waiver Wales wane waste wave waver way whale
See also quay.