So I've tried looking for some sources that make a numerical distinction between the casualties of British troops from the UK itself and those from British colonies who fought under their flag as well. Every source I've looked into so far only has the total of about 384,000 military deaths, all with a note that this includes the colonies as well.
As for the second part in the title, do we know how many British colonial troops volunteered vs. were conscripted? What was the reaction like in those colonies to men being conscripted?
If it's even possible, I think it'd be interesting to take a look at a breakdown by country, but anything that could point me in the right direction would be appreciated.
Greetings! Whilst I cannot unfortunately comment on the second half of your title and the exact number of volunteers for the colonies of the British Empire, I can shine some light on the distinction between casualties for the British troops and those who were recruited from the Empire. All of which come from R.A.C Parker's excellent publication The Second World War: A Short History. Let's begin.
The British Empire after 1945 had suffered irreversibly from the Second World War. From the Home Isles, the once mighty Britannia had lost 264,000 servicemen, and 90,000 civilians (most of whom perished during the Blitz of 1940-41). Across the empire, the white dominions had also suffered significant casualties:
The other British colonies combined (that is, those countries which were not self-governing dominions and had colonial ministries exercising control) lost 20,000 men. The British Raj in India had lost 32,000 men, but millions more are believed to have perished in the Bengal Famine of 1943, when crucial supplies of food in a drought were diverted to make up for the shortage of it elsewhere.
Hope this brief overview helps with the first part, and I look forward myself to seeing if any other contributor can weigh in on the second.
In addition to /u/Starwarsnerd222's excellent short summary, I have written about the reactions and responses of volunteers and conscripts to the Second World War in British East and West Africa within the larger context of the legacy of the First World War in Africa.