LW reception of Radio4 is unlikely to be very clear here - I'd say you might need good weather conditions to get good signal. Short Wave with BBC World Service might be an option for you?

I think the radio webcast is often only available to UK IP addresses; google public proxy list and choose a UK proxy and try streaming is a way around those limitations.

If you do bring your Sky box, I'd recommend leaving your dish and sourcing one locally of 80cm or more with a quad LNB. This means....you can get the absolute cheapest satbox you can find and use this with a hifi to get Radio 4 via satellite; possibly your best option. You should be able with practice to select radio stations "blind" and thus not need your TV to be on.