When using そばから in the sense of "as soon as", everything I find says that you can use either た form/past tense before it, or dictionary form, but not what the difference is. Sometimes people even use both forms in the same sentence:
我が家もそうなのだけれど、片付けるそばから散らかすし拭いたそばからこぼされる。
So is there any difference in nuance between these?
AそばからB is one of the many set phrases that connect two verbs A and B to express the sense of (near) simultaneity. Those phrases, while similar in meaning, have their own specific usages, grammatical restrictions, and connotations. For instance, やいなや always follows 連体形 (or the dictionary form/present tense if you will) of a verb while the perfect auxiliary verb た always precedes とたん (so, in short, it only takes the ta-form of a verb for A).
The important characteristics of そばから that set this phrase apart from others are that
verb A can be either present (the dictionary form) or with the perfect auxiliary verb た (ta-form) (or more like there is no restriction except semantics),
the described near simultaneous actions are typically being repeated and not a single event by one person, and
typically there is an implied sense that the speaker is negative, disapproving, or frustrated about the repeated near simultaneous actions.
A quintessential context is that whenever the speaker finishes task A, a person they know does the kind of thing that nullifies the completion of A, e.g., you tidy up a room, and lo and behold, your kid is already making a huge mess behind you. You clean up the room again, and your little one is about to drop food on the floor. Rinse and Repeat. Everyday. (And I think this is also the kind of context the example in the question is taken from.)
The choice of the verb form is up to the speaker; if you want to emphasize the connotation of repetition, the dictionary form is arguably better because it has this habitual semantics on its own, while inserting the perfect auxiliary verb (i.e., ta-form) would be more suited if you want to emphasize the fact that action B happens once action A has completed because the ta-form can add the sense of completion on its own.
I looked for information about whether there is a bias of some sort regarding which type is more frequently used for each verb, to no avail. Also, there seems to be my personal preference when it comes to some verbs. For instance, I seem to use 聞いたそばから忘れる far more often than 聞くそばから忘れる regardless of which sense (repetition or completion) I think is more important in a given situation. So I tend to resort to throwing in another word like いつも if I want to emphasize this sense even though this word is technically redundant. But I wouldn't say the latter (聞くそばから忘れる) is ungrammatical. And there seem many google hits for this type that seem to be written by native speakers, albeit somewhat fewer than the former.
The ている (or てる) version is the same. Some people seem to think this is not common. But this is not the case. So, for example, if you think the repeated chain actions are best described by this form because of the complete simultaneity between actions A and B, nothing prevents you from using this auxiliary verb, although your teacher may not like it if your textbook doesn't say it's ok; somehow no webpages for learners I just checked mentioned this version as a valid construction. But in reality, whatever they say, this version is prevalent and doesn't sound unnatural unless it is semantically unusual. If someone says you can't use ている/てる, just ask if "とか言ってるそばから自分で「〜してるそばから」って言っちゃったりしないですよね？" sounds unnatural. (It means "And you won't use '〜してるそばから' yourself like during this conversation, right?")
Oh, by the way, "〜" in the above sentence is pronounced 何々{なになに} if you're wondering.
From what I know, そばから is used to describe two events that take place soon after each other. Apparently, both the dictionary form and the た-form are perfectly acceptable, and the only difference is that た-form is used if the break between the two events is [subjectively] longer.
I personally have never seen/heard そばから accompanied by a verb in continuous tense (ている), but based on the native speakers' account, in this context (片付 takes a lot of time) the dictionary form and the continuous tense can be used interchangeably.
