Maybe if TV became popular earlier....(they had television in the 1920s BTW)
That's less trivial than it sounds, actually.
There was very early television from the late 1920s, yeah: W2XB in the Albany, NY area was licensed in 1928, and may be the first ever television station (the station, under its current call sign WRGB, claims this is the case). Regardless, it's hard to imagine this sort of television really taking flight, as it was about as primitive as you could get. Originally, it broadcast at 21 lines of resolution, later getting an upgrade to double that. That's less than 1/10th of a standard definition NTSC signal from today.
I know that the Washington, DC experimental station W3XK operated under similar limitations, and got around it (to some extent) by broadcasting only silhouettes. Add the expense and unreliability of early mechanical sets to the mix, and I can't see this kind of early television taking off so close to the Depression. At least, not in the USA.
If not for World War II, though, you might get television to launch a few years earlier (early 1940s instead of the late 1940s/early 1950s), since most of the required technology for what we'd recognize as modern television was in place by the late 1930s (and was on public display at the 1939 World's Fair, no less).
That said, I don't think you strictly need earlier television to push back the introduction of VCRs a bit. As it is, video tape was introduced in the early 1950s. The first enclosed
cassette didn't show up until the old U-matic things twenty some odd years later, but that seems like a relatively minor evolution once the initial work had been put into place.
The question, I guess, is
how much earlier? U-matic based VCRs were introduced to the consumer market in 1971. It's possible they could catch on in advance of the coming of VHS and Betamax in the latter half of the decade. Alternatively, I kind of like carlton_bach's suggestion of using early video tape as a kind of transport medium for television programming.
Also, how much should it resemble OTL VCRs? It's easier to imagine a
Quadruplex-based videotape-recording system in the early-to-mid-1960s than a earlier form of VHS. There are a few technical limitations of the format that prevented it from entering the consumer market, but my gut feeling is that all it really needs is some visionary to think of the consumer potential of the technology to speed things along considerably.