The simple fact is that many social welfare programs are just expensive, because there are a lot of people being helped and the quantity of money you need to spend to make any difference is just adds up.
When Medicare/Medicaid pays $150 for the same type of crutch that sells in a local pharmacy for $15, that difference does not go into "administrative expenses". I might prefer the expense of a human employee driving to Walgreens, buying a crutch off the shelf, and delivering it by hand to the patient that needs it, rather than paying a hospital that big markup just because that's the most they can charge without the benefits management program automatically denying the expense. Because even if it costs $135 in labor and transportation for a human to do all that, the mere possibility that might happen would discourage the hospital from charging significantly more than the pharmacy across the street for exactly the same item, or from charging different prices to different people based on their insurance plans.
It would even work to just give the patient $150 in cash, and tell them to buy a crutch with it and keep the change. Or carve one out of a tree branch and keep all of it. If they end up not having a crutch, it won't be because they couldn't afford it.
You can't rely on a bureaucracy's reports on itself to reveal the inefficiencies of that bureaucracy. There is just too much political pressure to cook the books.
They have 20 pages of corporate guidelines telling them how they have to lie. Zing!
Put another way, if the current welfare spending is £120 billion/year, the recipients are only getting £40-£60 billion/year. So that's all of the Basic Income you need to replace the welfare spending, leaving you with a £60-£80 billion/year surplus.
Obviously, the actual amounts depend on the detailed overhead costs for each agency that would be replaced by a UBI program. But I'd be willing to bet on anywhere from 20% to 60% savings for each agency. UBI can be highly automated; with electronic payments and ties to a tax database, it can be as little as a smallish office of administrators and DevOps, and a server farm.