I'm not sure that much could be done to improve the already pretty well equipped British military without hurting the already shaky civilian economy that was still coping with the Great Depression. The Commonwealth themselves I have no idea what else they could have done to increase production over OTL, but what effect did the commonwealth economic 'closed' market have on Commonwealth nations? It was probably cheaper to buy from Britain than to build up a new industry and compete, but I could be wrong.
But military re-armament with production at home would actually drive the local economy, and help in multiple ways. So cook up a deal so I don't know, Australia and either India or South Africa become production centers for the Commonwealth, with New Zealand building a lot of small arms, and everyone else pitching in resources, while Canada is to be more self-sufficient (aka leave them to their own devices,) but also a primary shipbuilder and producer of aircraft?
So thinking in starting from 1935-36:
Stuff like Hurricanes (Mk.1 Trops with 4x20mm would be best,) an improved Northrop A-17 mark, and the Blenheim/Beaufort and later Beaufighter for the Air Forces, and Wellington and Catalina for the Navy. Pretty much everything should use a Hercules or Merlin, to cut down on complexity.
For Surface vessels, go with simplified G-class Destroyers, and
Grimsby-class sloops for domestic production (build-able in Australia, later scaling up to Tribals and *
Bathurst-class,) and cheat beg and steal to get a few more cruisers or better yet, a Battleship and carrier, plus escorts assigned to Singapore (which also needs a better commander, and more and higher-quality seasoned troops, as if you're going to call some place the "Gibraltar of the East" you better damn treat it better than that.)
Also, engage in some horse trading to make damned sure the O/P/R-class Subs stay in the Pacific, along with HMS
Medway and another pair of Tenders.
As for the Reservists, the A9, A10, Universal Carrier, and a common armored car based off a *CMP 3-ton frame (with variants using the QF 2pdr and 20mm Oerlikon or HS.404,) are good choices for combat vehicles, and ought to be produced at home (and if possible, make the toolings to rapidly expand production and keep them in storage, just in case.)
Start building the *CMP immediately, and if pushed, say the frames on the older vehicles are too weak to last much longer, and say it's cheaper to replace them.
Start work on building the QF 2pdr and QF 6pdr, and start converting any QF 18prs to QF 25pdr Mk1s for Artillery, and start issuing 3-inch mortars and Boys AT Rifles (or better yet, Brownings, since they aren't crap,) to the regular and reservist units.
But for small arms, it's the Lee-Enfield Rifle No.1 Mk III* and Mk VI, in preparation for a new rifle, (refurbishing old ones and making enough so that all of the reservists have the Mk III* or Mk VI) a Submachine gun for MPs and Officers, either the Bren or .303 Madsen Gun (which ever is cheaper) as the squad machine gun, all for domestic production, supplementing the extant Lewis guns.
Ought to produce Vickers Ks and Browning M2s for Aircraft and Small Boats, (and make some light carriages for the Brownings, so they aren't hell on infantrymen using them as an Anti-tank or Anti-infantry weapon.)
As for the Troops, in Australia, Provide legislation requiring serious penalties for firing a member of the CMF for their participation in any Exercises, or other duties required of a Militiaman. Go to the one weekend a month, three weeks a year pattern after training, and offer a week's Active-duty pay for a weekend of service, and start recruiting, to restore the CMF to five Full-strength divisions, along with Active Duty units assigned to the various forts, and a few Brigades throughout the country.
New Zealand ought to similarly raise a division or two of Reservists, and the Royal Malay Regiment, Royal Hong Kong Regiment and SSVF ought to be brought up to the same standards of training and equipment.