I could see this having an effect on the ongoing
development of each language in a way that might leave
them closer together than they would otherwise be.
Yes, since Mandarin had been established as the Standard Chinese in the early 20th century, there definitely are effects that have ended up bringing Cantonese closer to Mandarin in terms of vocabulary and grammar.
For instance, if I take my previous example further, if such a situation existed with English and German, the words "wit" and "thou" might have ended up being more common in spoken English, or at least considered more formal rather than archaic. Certain grammatical constructions might also end up being more common, such as the continued use of the y- prefix in past participles.
I'm already stretching this analogy more that I should, but hopefully it gives a better idea of the relationship between Cantonese and Mandarin.
I also wanted to add a note that this effects how various Chinese speakers think of Chinese as a language.
Many Mandarin speakers see Mandarin as being equal to "the Chinese language" and often think of Cantonese as being "merely a dialect" of Mandarin because Cantonese speakers appear to read and write the way they do (even though the aforementioned examples show that this is really not the case), and they are not really exposed to the more major differences that would exist if Cantonese speakers actually wrote the way they spoke.
Cantonese speakers are more aware that they don't write the way they speak, but because the Chinese writing system only reflects the lexical and grammatical differences rather than the pronunciation differences (even though most of the differences are in the latter), many Cantonese speakers end up considering the written language as being a "formal" version of their language rather than a different language altogether. For this reason, many Cantonese speakers also consider Cantonese to be a dialect as well; however, they consider both Cantonese and Mandarin to be dialects of "Chinese" rather than Cantonese being a dialect of Mandarin.
Linguistically speaking, because spoken Cantonese and Mandarin are generally completely unintelligible to each other, they should be considered separate languages, but the the writing system ends up obscuring these differences. There are also political implications if Cantonese were considered a separate language because of Chinese nationalism tends to see that as a slippery slope towards separatism.
For instance, if I take my previous example further, if such a situation existed with English and German, the words "wit" and "thou" might have ended up being more common in spoken English, or at least considered more formal rather than archaic. Certain grammatical constructions might also end up being more common, such as the continued use of the y- prefix in past participles.
I'm already stretching this analogy more that I should, but hopefully it gives a better idea of the relationship between Cantonese and Mandarin.