1) Tires: Unfortunately, you can't get radials in 20" truck sizes. They don't exist.
Yes you can, they do exist, but you don't want them. When companies started making tubeless radials, they developed the half-inch convention. Radial tires are 19.5", 20.5", 22.5", etc. This was to prevent people from putting radials on the old tube-type, multiple-piece rims. I've never heard of, and have no manufacturer catalog references to 20.5" wheels/tires. Due to the bead difference, a radial (even of the same size as the rim) won't seat properly on the old rims and the tubes can't be used inside a radial to hold the air - then you get the exploding tire/tube issue. That being said, bias-plys are fine for this truck. Unless you change the running gear, you won't go fast enough to need radials and, besides, once the bias plys warm up the flat spots go away and they're a bit softer than the radials.
2) Converting to radial rims: While 19.5, xxxx, and 22.5" rims exist obviously, the real challenge is finding the correct ones to match Ford's five-bolt pattern if you stick to stock running gear. The 5 lug x 8" pattern was first used by Ford starting in 1934 and was used by almost all the medium duty truck manufacturers through the mid 1970s. It was heavily used on Ford and Dodge based motor homes. All the truck manufacturers sub'd out wheels to the same companies, i.e. Budd, Motor Wheel Corporation, Kelsey-Hayes, and later Accuride. The wheel manufacturers all used outer rims that were either made by, or at minimum designed and licensed by, Goodyear and Firestone. Ford never made these wheels. By the time radials and radial rims were available, Ford's not-so-common five-bolt pattern was long gone. I've heard International made rims briefly in the 1960s that fit the Ford pattern that were either 19.5" or 22.5" but I've never seen a set...this probably equates to the Holy Grail of Ford big truck afficianados. Actually no. Radials came on the truck market in the 1960s, but unfortunately tubeless truck wheels weren't able to handle the lateral loads imposed by these tires. I have a Budd Corporation SAE study that shows they had wheel failures attributed to radial tire usage. As a result, both Budd and Accuride, in their early catalogs, specifically state that warranties are voided if radials were mounted. After redesigning their products, the manufacturers coded wheels to show whether they were approved for radial tires. I have record of one 5 lug x 8" pattern 22.5" wheel, made by Accuride and listed in their 1982 catalog, that was radial approved. No Budd Company 22.5" was ever radial approved per my records. The 19.5" that was used heavily on motor homes is Budd #R89340. It is radial approved. These wheels are not real common, but do exist out there. I have several sets, and have one set available for sale.
3) Multi-piece rims: There are 2 types - the lock-ring-type and the 2-piece-type which have 2 halves that connect in the center Commonly called "widow makers", technically termed "Firestone RH-5°". (sometimes erroneously called "split rims" - split rims have the split across the tire and were never used on these trucks). Used on Dayton style demountable wheels, these rims are as dangerous as widow makers. The technical name for them is Goodyear "K" type. The lock-ring is the factory Ford rim I believe but aftermarket manufacturers (Budd, Firestone) made both the lock-ring and 2-piece rims. Dimensionally, they are the same and therefore interchangeable. I've got a mix of them on my truck. Take heed of the warnings when working with the rims but they aren't as deadly as people might lead you to believe. These rims are a good design and do not spontaneously explode if handled properly. Disagree, all can be deadly, but the lock ring style are generally safe once professionally assembled. Widow makers can and do explode spontaneously and should be pulled from service.
4) Tire sizes: COEs (and other big Ford trucks) had a 1.5-ton rating if the rear axle is single-speed and a 2-ton rating for the 2-speed axle (I'm not sure the 2-speed axle was an option in '38 or not). My truck has 7.50-20 tires as it has a 2-ton rating/2-speed axle. 1.5-ton trucks had either the 6.00-20, 6.50-20 or 7.00-20 tires over these years. The smaller the tire, the lower total load rating for the truck (the operator's manual for the heavy duty trucks tells you this number). For your purposes, I'd pick the best tire that matches your taste as load rating won't matter much. I personally prefer the 6.50- to 7.50-20s as they match the wheel openings better than the smaller tires.
5) Rim sizes: While all the big truck rims are 20" in diameter, some folks have already mentioned that rim width is a factor. Earlier trucks had narrower rims which limits the size of tire. If you want a big tire, you need the wider rim. Ford made 5, 6 and 7 inch rim widths (check the green bible parts book for correct year application - it will tell you the width). Offhand, I don't know the pairings but if you're interested, I can find out and get back to you.
Jeff and I have swapped notes on this already, but I'll go deeper into it. The late 1920s and 1930s were a time of rapid change for the wheel and tire manufacturers. During this era the National Wheel and Rim Association (NWRA) changed their method of recording tire sizes, and changed their method of measuring rim width. Specific to this discussion, they originally measured rim width across the rim from lip to lip. In the late 1930s or early 1940s, as best I can tell, they changed to measuring width across the inner rim at the base. Accordingly, what at the time was called a 5" rim, we today call a 3.75" rim. What was then a 6" rim is today called a 4.33" rim. During the 1940s the wheel manufacturers actually listed both these sizes side-by-side in the catalogs. By about 1950 they had completed the change over to the new NWRA standard.
6) Original running gear: I've got mine all original and it runs around town just fine. Remember, these COEs were designed for in-town delivery (the truck is about 3 feet shorter than the standard cab trucks and the steering design gave a very tight turning radius which was essential for maneuvering in tight alleys and loading docks). Consequently, long haul highway gears it doesn't have... If your running gear is in good shape, I'd keep it.
7) Interchangeability of parts with other Ford trucks: The COE mechanicals are largely interchangeable with all the other big (standard cab) Ford trucks. Not much is interchangeable with the car/pickup except the engine and ignition parts. Trim and body are different except for small stuff (door handles, locks, windshield wipers, etc.) Big trucks have the non-synchromesh 4-speed vice the car/pickup 3-speed. Rear axles are all different. Suspension is unique to the COE although some hardware can be shared with other big trucks (front and rear parallel springs did not agree with Henry but were a necessity on the COE with more cab/engine weight over the front axle).
8) Closed vs. open driveshaft: I don't recall if the '38-'39 COEs had a closed driveshaft or not. If they do, you can swap to a younger trans with the open driveshaft and rear axle. My '47 COE has an open driveline and that change may have occurred around '40 or '42...don't know...I'd have to research it. The 4-speed trans didn't change much except the rear mount and yoke assembly and whether there is a reverse lock-out lever on the shifter.
Hope you have fun with it...I know I have with mine! If I've got any erroneous data, someone please pipe up. Send me a PM anytime you have questions...glad to share what I know and always want to learn if I don't.
COE Dan