So, here is my non-exhaustive list of words familiar to most British citizens:
Drunk, pissed (two most common), inebriated, intoxicated, tipsy, hammered, bladdered, slaughtered, arseholed, smashed, under the influence, blind-drunk, sloshed, plastered, wasted, under the table, tight, three sheets to the wind, legless, merry, shit-faced, rat-arsed, half cut, paralytic, ratted, squiffy, wrecked, mullered, out of it, sozzled, boozed (up)
Tanked, potted, stewed, sauced, laced, totalled, leathered, lathered, addled, canned, soused, welly'd, pie-eyed, steaming, pickled, blotto, stocious
OK, you may disagree with me on which ones are common or less common, and much depends on social class, age and region.
I'm afraid I cannot go into the finer points of the French synonyms for "ivre", but I did find this list of 44 words and phrases. Here we go:
Ivre, soûl/saoul (comme un Polonais), bourré, beurré, enivré, éméché, paf, brindezingue, aviné, fou, en goguette, imbriaque, pompette, émoustillé, cané, gai, mûr, pété, rond (comme un piron), ému, bituré, cuit, cuité, enragé, gris, mort, pinté, schlass, blindé, éperdu, exalté, grisé, pâmé, plein, avide, en brosse, fanatique, hourdé, parti, poivré, pompette, transporté, torchon, carpet
I have to confess that I only recognised five of those. I should read more. Thanks to Sylvie Rawlings for paf and pompette, Aurélie Hill for torchon and carpet and Alex Bellars for déchiré.