If it gums up the pawls, it will be a zero speed.

3-in-One Motor Oil (blue label) is a non-detergent 20wt. oil and is probably the closest modern equivalent to the old Sturmey-Archer Cycle Oil. It even comes in a similar dispenser bottle.

3-in-One Multi-Purpose Oil (black label) debuted during the first great bike boom in 1894, making it one of the oldest cycling products you can still buy. The oil was originally intended for bicycle chains, and the name indicated it "1) cleaned, 2) lubricated and 3) rust proofed", hence, 3-in-One. After 119 years, it's still not a bad choice for chain lubrication.

3-in-One Multipurpose Oil contains a vegetable based component, citronella oil (ever notice the way 3-in-One Multipurpose smells?), which will go rancid, break down and turn into very much a non-lubricant. This residue would get cleaned off a chain in the next application, but when enclosed in a small metal shell it has nowhere to go. Probably more Sturmeys in the USA have been rendered inoperable by 3-in-One Multi-Purpose residue than for any other reason. The 3-in-One folks themselves do not list IGHs as a potential use for their Multipurpose Oil.