Hi Alex,
It depends on what your main goals are. Seeing as it's your first trip to the Western states, you'll have absolutely plenty to keep you busy at any time of year, and thankfully SE Arizona is blessed with sensational year round birding. :t:
Late April/early May is usually a very pleasant time to be here. The birding is excellent and although still hotter than your average Scottish mid summer, it's something you can bird in quite comfortably. It's a popular time to visit for many birders, with spring migration in full swing and some of the rare Mexican summer visitors arriving.
Late August - like it is right now - can be really hot and humid, with the monsoon season rumbling on as it is this year. If you can deal with the heat it's a fantastic and truly exciting time to be here. Plus you will have the option of seeing more of the specialist SE Arizona species, some of which don't arrive until mid to late May. If you have set target species, this could be critical. For example:
Some of the hummingbirds, such as White-eared and Berylline, aren't always regular until midway through May.
Cordilleran Flycatcher, Brown-crested Flycatcher and Thick-billed Kingbird don't turn up until early May, Tropical Kingbird mid May, Sulphur-bellied Flycatcher late May/early June. Traditionally Rose-throated Becards don't arrive until early to mid May, but sadly none have been seen in SE Arizona in summer since 2007 (a few winter records though).
Botteri's Sparrow isn't usually seen before mid May, and even then it's difficult to detect until they sing in the monsoon season, July & August.
Two real stunners, Blue Grosbeak and Varied Bunting, aren't seen until late May or June.
If those would be important to you, it has to be late summer. Otherwise, spring will probably be your best bet. If you have the option on dates, make it more towards early May rather than late April to maximise your summer migrants, again if that's more important than pleasant weather.
Whatever time you come here, you should have a fantastic trip!