In short, yes, it would be correct to say that the British Empire had a Scottish or (to a lesser degree) Welsh character. Scots were, indeed, greatly overrepresented in many colonial endeavors.

The British Empire was many things at many different times and in many different places, so for the sake of simplicity I'll just use two examples, mainly dealing with Scottish people. Scots were massively overrepresented as immigrants in Canada early on. Parliament paid for the importation of 2000 Glaswegians to Upper Canada (Ontario) in the early 19th century while over 17,000 Scottish Gaelic speakers had settled in Nova Scotia by 1812 (about 1% of Scotland's total population at that time). As late as 1871, the Scottish born population of Canada (125,450) was nearly at parity with the English and Welsh born population (147,081). Even by 1961, about 30% of British born Canadians were born in Scotland... a massive over-representation considering that Scots make up only 8% of the UK's population! Both of Canada's first two Prime Ministers were born in Scotland as was William Lyon Mackenzie, the first mayor of Toronto and the grandfather of the country's longest serving PM.

Now, you may be thinking that this isn't what you meant... after all, settlement in a British colony was just one mode of imperialism. You may have been thinking of a more commonly expressed image, men in pith helmets sipping tea in India. But even there, Scots were massively overrepresented. In the late 18th century, Scots represented nearly 50% of East India Company writers and officers in Bengal, over 50% of the EIC's medical recruits, and 60% of all traders and merchants licensed by the Company to operate in India. They were proportionally fewer when the EIC gave way to crown control and appointments to the Indian Civil Service were made by exam, however Scots continued to be largely overrepresented in private industry. Scottish businessmen dominated the Indian jute and oil industries, for example. Since you asked about the aristocracy in particular, it's notable that the colonial services were much more middle class than you'd probably imagine. However, Scots did tend to send more of their landed gentry overseas than the English.

As for the Welsh, David Gilmour's magisterial The British in India quotes a Scottish historian as saying "the Welsh failed to make their mark in India..." and that is true. There were of course notable exceptions, Sir George Everest certainly left his name on the subcontinent if nothing else, but Welsh activity was largely restricted to Methodist missions. The same is true in Canada, in Canada and the British Empire Marjory Harper says "the lack of scholarly attention paid to Welsh migrants in Canada is attributable to their statistical anonymity, rapid assimilation, and preference for dispersed over clustered settlement."

This is really only scratching the surface of this topic. It doesn't touch on, for example, Irish Protestants (and even nationalist Irish Catholics, in some cases) who were also statistically over-represented both as settler colonialists and colonial officials at various times throughout the Empire's history. But suffice to say, the meme is indeed well-grounded.

Sources:

The British in India: A Social History of the Raj by David Gilmour

The Oxford History of the British Empire: Canada and the British Empire edited by Philip Buckner