I've done it a few times from London. Bus from Leon to Oviedo bus station (quite a lot, and quite regular), then another from Oviedo bus station to Asturias airport (very frequent, cheap and about half an hour). Or, train from Leon to Santiago, and then a bus to Santiago airport. Or train/bus from Leon to Madrid.
I always work it out depending on the cheapest flight I can find. I usually pay between £20 - 40 using Ryan Air, Easy Jet, Vueling or Norwegian. All budget, all let you take a backpack as carry-on luggage.
I use Skyscanner which gives a comprehensive run down of flights.