Channel 36, within Europe's (and most of the world's) 8MHz spacing) is positioned at 591.25 MHz (video carrier) which is the same frequency as channel 34 in the US with its 6 Mhz spacing (*1). Thus the same modulator (circuit/unit) can be used in European and US devices, simplifying production - and making it a cheap part to buy for a product needing TV output in the mid 1980s.
How were UHF channels allocated to home computers?
Not at all, as it was a decision made by manufacturers, not done by any government/international agency.
most (all?) home computers with UHF TV outputs used channel 36,
Not really, it only started in the early '80s, most notably with British machines.
along with VCRs
Which resulted in extreme high volume production, making channel 36 modulators dirt cheap.
Is it coincidental that this was an adjacent channel to the empty one?
True, as that's only true for the numerically close US channel. Which only happened by accident due to the way they are numbered (see below).
How was this allocation determined?
By looking at frequency assignment tables for countries all around the word.
Some Fine Print
Since US (and thus most of the Americas) and Europe (most of the remaining planet) both start their UHF assignments at 471.25 MHz (*2) meaning that each third European channel is at the same frequency as each fourth US channel. In addition, Europe called the first channel 21, while the US started its numbering with 14. As a result the following channel 'pairs' would have been available:
21/14 471.25 Band IV
--> 36/34 591.25 BAND V
42/42 639.25 + Australia 44
So any of these would have worked fine all around the world. Further, in 1961 UHF for broadcast was defined in two 'Bands': Band IV up to 582 MHz and Band V over 582 MHz. So 591.25 is
- the first common carrier frequency for Europe and the US,
- in both schemes the second Band V channel,
- the last channel below 600 MHz
- far from early assigned channels in all countries as assignment started low.
- likely tunable by all UHF capable TV sets, even older ones made for Band IV only.
So 591.25 is simply a sweet spot, screaming to be selected.
On a side note:
The Aussies got left out again. Their spacing not only starts at 520,25 MHz, but uses a 7 MHz spacing, and starts numbering at 27, making their channel 28 the same as Europe's 28. So while a 3:4 relation resolves to a high repetition rate, a 6:7:8 relation gives only a single channel in all of assigned UHF common to all three systems:
639.25 MHz Aus 44 - US 42 - Europe 42 (*3)
Quite handy, except far into Band V, above 600 MHz and outside older TV. Then again, 591.25 is 'only' off by one MHz from Australian channel 37 at 590.25, so even their TVs should be able to fine tune.
On a related comment by Kaz
Wouldn't you still need a different modulator unit, given the different TV standards (NTSC/PAL)?
No, Modulators are about putting a signal on a carrier, they are not part of a TV signal generation, which happens before, but part of signal transmission. A modulator takes a signal and modulates it on top of whatever frequency it is set for. It doesn't care about that signal structure, as long as it's within a given spectrum (and anything outside will only add artefacts). So a modulator capable of putting an 8 MHz signal onto an UHF channel, will work fine for any TV standard that doesn't exceed 8 MHz baseband.
Of course there might be some naming confusion when it comes to home computers. The term modulator describes only a unit with above function, but there were boxes for some home computers that did more than lifting a TV signal to transmission frequency. For example when the signal was delivered as components (Colour, Intensity and Sound), then it had first to mix and move each component into a standard-conforming baseband TV signal first before modulating it onto a channel.
These devices include more components than just the modulator - even though post companies/users simply called it that.
*1 - And channel 33 in Japan, as they used the same spectrum and width as the US, but started their UHF numbering with 13.
*2 - Video Carrier Frequency. The channels start at 470, but it's more useful to compare the carrier as that's the one modulators and demodulators have to be tuned for.
*3 - Sure, we know it's the Answer, still, it looks suspicious, as it's also the only channel where number and frequency is the same in both systems. As if it was on purpose - no, I do not know any further information, I'm just caught by the numbers.