The Australian-English phrase (as) dry as a Pommy’s towel, and its variants, mean very dry.

The Australian-English noun Pommy, also Pommie, and the shortened form Pom, designate:
– a British immigrant to Australia;
– a British (especially an English) person.

In the phrase (as) dry as a Pommy’s towel and variants, the allusion is to the alleged poor personal hygiene of the British—cf., below, quotations 4, 5, 6 and 7.

The earliest occurrences of the phrase dry as a Pommy’s towel and variants that I have found are as follows, in chronological order:

1-: From The National Times: Australia’s National Weekly of Business and Affairs (Sydney, New South Wales) of Sunday 4^th January 1981—as quoted in A Dictionary of Australian Colloquialisms (Sydney University Press in association with Oxford University Press Australia, 1990), by Gerald Alfred Wilkes (1927-2020):

Humphries had McKenzie downing his first beer and exclaiming he had been ‘as dry as a Pommie’s towel’.

2-: From Sam White Down Under, published in The New Standard (London, England) of Friday 20^th February 1981:

Still no peace for the Poms

[…]
It is ironic to note that while old prejudices are dying, they seem to be dying hardest in relation to the Poms. There was always a strong anti-British element in Australia stemming partly from its history, but largely from its large Irish population and from a burgeoning Australian nationalism which resented what it saw as Australia’s semi-colonial status.
Now however it is the conservative elements in the community who were once the keenest to maintain the British connection who complain most about the British migrants, blaming them for a militancy in the Australian trade unions which has in fact always existed and is purely native in origin.
By contrast popular anti-Pommy sentiment still reflects the country’s growing nationalism—one might even call it a nascent Republicanism, which often is expressed in crude but not unfunny jibes such as describing a parched piece of land as being “drier than a Pommy’s towel”.

3-: From Some hot times in Adelaide—the land of the long white sock, by Leo Schofield, reporting from the Adelaide Festival, published in The Sydney Morning Herald (Sydney, New South Wales) of Saturday 13^th March 1982:

It was dry as a Pommy’s bathmat in Adelaide last week. Dry and stinking hot. Moving from plane to waiting car, one felt like a turkey in a convection oven.

4-: From Super sport cricket special: Mighty mouth, by Karl Kershaw, published in the Sunday People (London, England) of Sunday 7^th November 1982:

The grossly beer-bellied character in an Australian TV commercial wipes his sweaty brow and gasps “Streuth, I’m as dry as a Pommie’s towel!”
That’s just a mild example of sledging and sustains the Aussie myth that the English do not take enough baths.

5-: From The Awful Aussie: myth-making in Fleet Street, by Margaret Jones, published in The Sydney Morning Herald (Sydney, New South Wales) of Saturday 19^th March 1983:

London, Friday.—On the eve of the Royal tour, Australians resident in Britain are bracing themselves for a wave of Awful Aussie jokes.
The London borough of Islington recently banned Irish jokes, on the grounds that they are racist. A ban on Down Under jokes might be a good idea. Though the Irish are usually portrayed as bumbling incompetents, they still do better than the natives of the “land of Fosters Lagers, blowflies, and the Vegemite sandwich.”
That quote comes from the London Sun, which also runs a special guide on “Aussiespeak,” featuring such classic Australianisms as “Ow ya going, orright?” (“How do you do”) and “Wanna squirt of Bushells?” (“Would you care for a cup of tea?”).
The Princess of Wales is warned that when she arrives in Australia, she will be sprayed all over by a man with a big aerosol can, just to make sure she brings no English germs into the country.
The Sun recommends that as Australians are convinced Pommies never wash, she should at least wet the bathroom towels to placate her hosts. (Hence the expression, quoted in the Aussiespeak guide: “Dry as a Pommy’s towel” (“Longing for a tinny”.)

6-: From The Sunday Carlton Report, by Mike Carlton, published in The Sun-Herald (Sydney, New South Wales) of Sunday 20^th March 1983:

It beats me that the British simply cannot get Australia right. After all, we’ve been around long enough, and we certainly know all about them.
It is perfectly understood here that Poms eat nothing but whelks and jellied eels, do not wash, live in thatched cottages and keep the coal in the bath. The saying “as dry as a Pommy’s towel” has become part of the language.

7-: From A Pom is a Pom, but gays aren’t always gay, by Margaret Jones, published in The Sydney Morning Herald (Sydney, New South Wales) of Friday 4^th May 1984:

What is a Pom, anyway? That’s an easy one. Poms by their very nature are pale, adenoidal and whingeing, and don’t bath as often as they should (hence the saying: “As dry as a Pommy’s towel”). They are either upper class and insufferably patronising, or aggressively working class with funny accents, and carriers of the British Disease.
Poms have strange rituals like standing in queues and drinking lukewarm beer, as every Ocker who comes back from Pommyland will tell you. Ockers, in fact, are not all that much impressed by Britain. (“Did y’ have a good time, Bruce?” “Yeah, it was all right except for the Poms.”)