If you're taking your bike, you can cycle along a disused line from Bishop Auckland to Durham.
All the cheap tickets have sold out, probably a while ago. It's a popular route.
If you have a railcard (if you don't, get one!), GC is not going to cost more than the plane, at £46.85 to Sunderland or £44.20 to Eaglescliffe. A single from Eaglescliffe to Bishop Auckland is £3.55 with a Railcard. All these tickets can be bought on board, and the off peak ticket is valid on the 0804 from London, and any train back during the weekend. The 1730 from Sunderland may suit your needs.
If you're up for cycling some of the many disused lines while you're in the area that can be arranged! (The
Waskerley Way is highly recommended)
Is the plane fare really £46 return including everything at such short notice? If it takes you to Newcastle Airport, you'll need tickets for the Metro + Newcastle to Bishop Auckland / Durham (for which no "open" return is available so you'd need to buy 2 x singles) so even if the plane is only £46 it will cost more than GC.
Still a good chance of getting Valentas on GC too, so get them while you can! Taking a bike on board GC is no problem at all. You may get to meet BOSCH and/or Sir_Clagalot.
Edit: I found a cheap ticket going north, for £14.50 without a railcard with this itinerary, specifying via Leeds:
KGX 0635
LDS 0850
LDS 0927
DAR 1032
DAR 1055
BIA 1121
For the way back, the Advance
singles cost more than a walk-on Off Peak
return in some cases! And First Class is cheaper than Standard class on most trains!
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Looking at plane fares..
Eazyjet's website gives a fare of £31 there (Dep Stansted 0700) and between £25 and £35 back however there are "No flights available" according to their website on the evening of 08 May, you'd need to return on Sunday 09 May instead (£35 on the 1720). There are NO BAGS allowed at that price, if you wish to take a bag and come back Sunday then the total will be £31 + £35 + £18 = £84 Plus you need to get to Stanstead. You'd need an Anytime ticket, an SOR from London Terminals (you'd need to get there first!) is £28.70 without a Railcard or £18.95 with a Railcard. Then add the Tyne & Wear Metro plus 2 x singles from Newcastle to Durham / Bishop Auckland. The total to go by DifficultJet is likely to be in the region of £146, not £46. Oh, and that is excluding insurance. Eazyjet admit it is a "risk" to fly on their planes without it!
BA is cheaper than DifficultJet, as the fare is only £10 more once you take a bag, and you can get to Heathrow cheaper on Oyster.
So I'd go with GC!
Oh, and GC won't be affected by the "wrong type of ash"!