The name is not to be found in the Oxford Names Companion, but its etymology would appear to be pretty straightforward. Just as Upjohn is of Welsh origin, a patronymic from John with the prefix ap ‘son of’, so Uprichard must presumably be from Richard with the same prefix. In modern Welsh it would accordingly appear as ap Rhisiart.
The ju- pronunciation is then to be explained as arising from the spelling: compare utility, ukulele, upas, Urals.
I don’t know what the etymology of the well-known novelist John Updike’s name is: but it looks as though that would be quite different, being of Germanic (= English) origin, ‘upper ditch/dyke’.
The Welsh for ‘son’ is mab maːb, corresponding to q-Celtic Mac, Mc-. In English it can be reduced to b-, as in Bevan, Beynon, Bowen, and some cases of Barry and Beaumont; or to p-, as in Parry, Pugh (ap Huw), Pomphrey/Pumphrey, Powell (ap Hywel), Preece/Price (ap Rhys), Probert, Prothero (ap Rhydderch), presumably Prodger and indeed Pritchard. Upjohn and Uprichard (?) seem to be the only cases in which it surfaces in English as Up-.
For the counteretymological spelling pronunciation of initial U- as ju we can compare the placename Uttoxeter, Staffs, which can be juˈtɒksɪtə or ʌˈtɒksɪtə, or even ˈʌksɪtə. It appears in the Domesday Book (1086) as Wotocheshede and may mean ‘Wuttuc’s heath’.
There's also Udimore in Sussex, which can be ˈjuːdɪmɔː or ˈʌdɪmɔː.