As you say, pretty much all issues around the legacy of colonialism are contested, but my view is that it’s impossible to make a generalisation about British institutions versus others. Pretty much all my information is based on sub-Saharan Africa, so I’m not certain how different a comparison between, say, British Malaya and the Dutch East Indies would be, but I imagine the general argument would remain the same.

As mentioned already, its common for historians to contrast the French and British approaches to their colonies: according to this, the French exported their centralised prefectorial system which led to more centralised independent states, whereas British ‘indirect rule’ led to more decentralised states. In theory, this could lead to ‘better’ institutions in British colonies because local rulers could tailor them to the local situation, which was especially helpful in an African context where states often had fairly arbitrary borders incorporating many different groups.

However, British and French rule on the ground was not as different as the official ideologies of imperialism suggest: in practice there were usually far too few Europeans in any of the colonies to run a true system of ‘direct rule’ on the French model, and all colonial systems were improvised based on local situations. Increasingly, many political scientists now emphasise longer-run influences on institution-formation, which can more effectively account for differences between and within countries. Cathy Boone, for example, argues that institutions were developed through ‘strategic bargaining’ between social groups in different parts of each colony, which led to different types of institution based on local social structures and economic interests.

To take a British example: in Uganda, the institutions developed in the historic kingdom of Buganda, which was home to a fairly centralised pre-colonial state, were different to those in Teso district, where Iteso society was more egalitarian and lacked an established bureaucracy. In Buganda the British were pretty happy to let the Kabaka – the local king – keep most of his power and rule in the same way as previously, provided he remained loyal to the British. In Teso, in contrast, the colonial administration invented a whole new system of chiefdoms, with the chiefs paid through a rebate from taxes they collected for the central government. In West Nile, meanwhile, the British were far less interested, and created basically no new institutions at all.

Basically, while it may be that in some cases the institutions left behind in certain parts of British colonies were more effective, this was most likely the result of local socioeconomic factors, not anything inherent in the British system of colonialism.

Sources

For the argument that does see a difference between different nations' systems of colonialism, see William Miles (1994), Hausaland Divided: Colonialism and Independence in Nigeria and Niger

For the alternative, which I mostly agree with, see Catherine Boone (2003), Political Topographies of the African State: Territorial Authority and Institutional Choice

I can say that "permanent" British colonization of Kenya was better for the country than Britain's temporary guardianship of Tanzania was for that country. In the former, the Brits invested in large amounts of infrastructure for long term extraction of resources, while it knew that it was scheduled to leave Tanzania on a certain timeframe and so just concentrated on extracting as much as possible on whatever the Germans had left behind.

The Brits also were fairly happy to allow a certain level of self-rule as long as quotas were met, while other colonial powers were more hands-on.

This is a relatively small part of the overall story but that policing and intelligence a function of civilian government often preserved the liberties under corrupt and authoritarian regimes to a greater extent than the post-colonial regimes where law enforcement and intelligence gathering had typically been the preserve of the military.

Counter-espionage and counter-terrorism was the preserve of the police Special Branch, which meant new governments already had an organisation in place that fulfilled the remit others would use secret police or party militia for. Similarly, while nations were part of the Empire, they fell under the remit of MI5 (the Secret Service) rather than MI6 (Secret Intelligence Service), and on the eve of independence the security apparatus would be turned over to the new government, usually in a secret meeting that made reams of dossiers available.

They then suggested that the new government retain the existing heads of station as a Security Liaison Officer (SLO) etc for their tremendous experience (and secret dossiers on their political rivals, which despite being a target of himself, India's first PM Jawaharlal Nehru was happy to take advantage of). This was, of course, to allow Britain to continue its intelligence-gathering in the country despite having lowered the flag and boxed up the whiskey in Government House, but it ensured that domestic intelligence, in the short term at least, remained at a distance from the often volatile politics of a new state. Similarly, veteran Special Branch officers were also retained for their wealth of experience, while many of the existing officers would have been a product of British police training.

Obviously this isn't a totally impeachable safeguard against tyranny, and in some cases the existing authoritarian disposition of stay-behinds played a role, but both the inheritance of existing organisations (and in some cases existing staff) ensured that the most eagerly abused tools of any new state enjoyed a constitutional framework, with experienced officers and oversight, and separate from the military or political parties.

Which isn't to say these institutions couldn't be used for repression either sides of independence (the Mau Mau Insurgency in Kenya and the bungled counter-insurgency in Palestine are particularly grim examples of the former), but there are examples I've read of Special Branch or MI5 actively undermining newly installed autocrats - including at least one of them helping the moderate opposition oust them from power. I'll have to check the exact scenario when I get home as I don't have my references to hand.

For an example of the ethical pangs that British intelligence occasionally suffered, after assisting Australia in setting up their own agencies in 1948/9, they refused to do the same for South Africa with MI5 Director General Sir Percy Sillitoe visiting the country and reporting back to British PM Clement Attlee that:

"The improper uses to which a Security Service might be put by the Nationalists might well include its employment against the Parliamentary Opposition and against those members of the British community out of sympathy with the Nationalist political programme. It would certainly be used to keep down the black races."

TL;DR: A lot of the time the British turned over a decent domestic intelligence network and a decent counter-terrorism/espionage unit with civilian roots so would be dictators were less likely to create a new secret paramilitary death squad to do the job.

SOURCES: Empire of Secrets: British Intelligence, the Cold War and the Twilight of Empire by Calder Walton and The Defence of the Realm: The Authorized History of MI5 by Christopher Andrew.