While Danny and Oll are right on this, you probably should pay it now and then kick up a massive stink that they haven't delivered it.
THAT SAID
They will not refund the customs charge - they will refer you to the custom service who'll refer you back to royal mail who'll refer you to customs service until either (a) you die of boredom or (b) hell freezes over and they admit fault. Customs will stand behind "we're a collector of charges for HM, after that we tell royal mail to deliver it, as far as we're concerned it's in the UK and not in our hands - we never physically see the parcel after it passes through security, we have the carrier hold it" (this is true, parcels come in, get cleared by security are tagged as revenue collectable and then taken onwards - usually FedEx or UPS will pay the charge and then collect it from you, i've had a couple like this from Taiwan. If you don't pay or they can't contact you the parcel is generally returned and the customs invoice to the courier is negated - they all work on 90 day payments so the idea being that by the time its due the recipient has paid it so nobody's out of pocket).
Next, RM will tell you they're "Very sorry for your problems with...." in a template letter and that ".... we're looking into it, please bear with us, this will take us about 30 days....".
At which point they'll say that ".... international parcels cannot be tracked after leaving their originating country" even though there's an international standard for the 13 digit tracking number designed exactly for this according to their own website and "we've conducted a thorough search" and finally "... there's nothing we can do at this point, you should contact the originator and claim from their postal service".
A little story...
I had a problem with a parcel from Scott Brassington - I still haven't received the goods btw and this was from May last year but we're in touch, it's not Scott's fault.
Ordered beginning of May, Scott shipped within days, got a card about 10 days later saying I had a customs charge + the £8 handling fee, paid online immediately. 1 week, no parcel, 2 weeks no parcel.
Rang Royal Mail who I had to argue for 10 mins with their stupid voice input system at 8p/min to be told by someone that they had no idea where the parcel was because it wasn't a tracked service but could have my post office redeliver it on Friday - so its at the post office, no sir its not a tracked service I can't tell you that (went to post office, all layabout workers stood around denying it was there) - one left me stood in the delivery office for 30 mins while he went outside for a fag. I nearly got escorted off the site for making a complaint about that.
Still no redelivery on the date scheduled or the following Monday, rang delivery office, nobody answers, went in, told can't be helped without a non-delivery card please move along sir there are other people waiting.
Fast forward about 10 weeks and I get an email from Scott with a photo saying the parcel had been returned to him - it had writing on it saying "redeliver 24th" and then a sticker saying "never collected".
I took this to the post office, standard unionistic "we can't help you, call this number" rubbish, nobody knows anything, we're underpaid and overworked blah blah, I even still had the original customs card (but miraculously there's no parcel reference number on it).
So, at the moment, i'm out £120+ for the items, £12 for parts, about 2 days in time and still no parts.
Scott has them but we daren't ship them again because it'll just repeat itself.
Its with the Postal Redress Service but Royal Fail are just stuck with their feet in the mud saying "it wasn't a tracked service" and refusing to admit they never tried to deliver it.
At one point you paid £1.45 for a recorded delivery signed for letter, you can still pay that but the workers in the post office just continually bleat "it's not a tracked service" upselling you - you just know that if you don't pay the extra it'll just go in the shredder and they'll just send you some stamps as compensation.
Btw, more recently my partner sent a parcel using "express 24" at £20 to send a parcel to her daughter in Scotland just before Christmas - they must have missed a digit as it seems like "express 240". It was a guaranteed service and yet took 7 days to go to the local hub office, 1 to go up to the sorting hub in Preston, 2 more to their local area hub and then finally another 2 to be delivered, the grandson got his Christmas presents on 2nd Jan.
Bit of a long post but my opinion now is play them at their own game - screw them over if you possibly can.