Epistemic modality ("He will be in London now") and futurity ("He will see her tomorrow") are generally associated with each other. This seems to be fairly common cross-linguistically, but there are enough subtle differences between them to make it quite individual to the speech variety in question.
In Old English, no synthetic future tense is attested, and future meanings are usually expressed with present tense verbs.
However we do see hints of some kind of future meaning derived from both "sculan", a modal verb meaning "must/should" and a transitive main verb meaning "owe"; and "willan", the modal and transitive main verb meaning "want".
E.g. from the years 996-7, Ælfric's Lives of Saints, "Of Saint Maurus", line 327 (or 330):
and þæt þæt he þe sæde is soð be dæle, swa þæt se mæste dæl ðinre muneca sceal of life gewitan binnan lytlan fyrste.
and that that he to-thee said is true in part, viz that the greatest part of-your monks shall from life go within little time.
and what he said is partly true, that is, that most of your monks will die within a short time.
There is a link drawn in Traugott (1989), to the Latin future active participle, in -turus. But Traugott goes on to say:
All these and other 'future'-oriented examples of OE willan and *sculan involve either a generalization, the 'prophetic/obligated' future, or simply 'later time' (relative tense). They are not deictic futures in which the time of utterance is the reference point. In other words, they are not fully subjective tenses dependent for the interpretation on knowledge of speaker time.
It was in Middle English that the "intentional subjective future" became strong grammaticalised, e.g. from 1386 Chaucer's Canterbury Tales:
I wol gladly yelden hire my place.
I will gladly yield my position to her.
Epistemic modality seems to have emerged a bit earlier, to my eyes at least. Traugott cites the following from version E of the Anglo-Saxon Chronicle, dated to 755:
swa swa mænige sæden þe hit geseon sceoldan.
as (=) many said that it see should.
as many who supposedly saw it (must have seen it) said.
However, the subjective version, where the use of the construction "shall/will + infinitive" implies the speaker's supposition, is much later. Visser (1966) dates it to the 19th century in Standard English, with earlier occurrences limited to Scottish and Northern English, and states that even as late as 1903, it was not wholly accepted.
In Italian (as well as Spanish and Modern Greek), there is a use of the synthetic "future tense" in an epistemic role where English would take the must modal verb + infinitive construction, or the plain "present simple":
Non è a scuola. Sarà malato.
not is at school. be-FUT.3S ill
He is not at school. He must be ill.
Interestingly, French behaves like English, using the present of devoir + infinitive in this case. One of the proposed synchronic explanations of the difference is the future ratification hypothesis. Hence, we can see how there are differences in the precise nature of the epistemic modality covered by each construction in these languages, providing evidence that these developments are at root independent.
Senti's 2015 study on medieval, 15th- and 16th-century Catalan also shows how the division of labour between epistemic modality ("inference"), future tense, and deontic modality ("necessity") for the verbs deure and haver changed quite dramatically between the centuries. In Modern Catalan, deure has pretty much completely lost its temporal futurity. Again, each individual language's tense system has developed differently.
The case of Modern Korean 겠 (Yale: geyss) is very interesting, as it is a "future" that still retains a strong flavour of both deontic and epistemic modality. Its emergence dates from the transition of Middle Korean to Early Modern Korean in the 16th century, emerging from the merger of -게 ᄒᆞ- (-gey ho-), which was a causative [now taken over by -게 되다] then a passive, and anteriority/perfective marker -였- (-yess-); the product of this gained a meaning of "definite imposition of obligation". From that field of deontic modality, the epistemic modality and futurity meanings appeared. Of course, there are many other future-time constructions in Korean, and the specific shades of modality, futurity, voltion and assertion (as well as limits of politeness/formality levels) are all entangled.
Hence, these concepts are intimately related, whether it's in English, in Tundra Nenets, or in Mandarin Chinese. It's interesting to see what constructions will fill the role of futurity and epistemic and deontic modalities, and in what ways they'll continue to develop.