On the 9th, £20.75 on the 08:35 or 10:35, with a change in Oxford. Straightforward.
On the 27th:
Itinerary:
Didcot - Oxford: First Great Western train 07:31, arr. 07:49
Oxford - Banbury: bus replacement 07:55, arr. 08:45
Banbury - Birmingham New Street - Cross Country train 08:54, arr. 09:48
Birmingham New Street - Newcastle Cross Country train, 10:30 arr. 13:45
Tickets:
Didcot - Banbury £4.25 Offpeak day single
Banbury - Birmingham New Street £4.25 Advance single (08:54 train)
Birmingham New Street - York £13 Advance (10:30 train)
York - Newcastle £6.25 Advance Advance (12:48 train)
Total: £27.75
Note that the Birmingham - Newcastle train is split into two advances on the same train. I'm not sure how reliable the scheduling is on the Oxford - Banbury bus replacement, but in theory you have 9 minutes, which is a valid connection. If you missed the train in Banbury, you'd board the 09:08 to Birmingham Moor Street (Chiltern Trains) instead, and then walk across to Birmingham New Street.
If you prefer to eliminate the bustitution, then:
07:12 from Didcot to Bristol Temple Meads arr. 08:17
08:30 from Bristol Temple Meads to Birmingham New Street, arr. 09:56
Then the same tickets/train between Birmingham and Newcastle as before.
Tickets: Didcot - Bristol Temple Meads £3.90 Advance (07:12)
Bristol Temple Meads - Cheltenham Spa £4.20 Off-Peak Day Single (or book an Advance for £4 if you really must, 08:30)
Cheltenham Spa - Birmingham New Street £5.50 Advance (09:12)
Total: £32.85 to Newcastle
It is possible to make this journey later also. There is a Birmingham - Newcastle Advance fare of £18.50 on the 17:03 and 19:30, and the £19.25 split at York is available every hour from 10:30 onwards, so you don't have to leave Didcot first thing if you don't want to. But around £30 is a good fare.
Easier: drive/taxi to Oxford then the Megabus direct to Newcastle.
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Second thoughts:
easiest option, two trains:
Didcot Parkway - Bristol Parkway 14:22 arrives 15:06
Bristol Parkway - Newcastle 15:40, arrives 20:33
Ticketing:
Didcot - Cheltenham Spa Super Off Peak Single Route: Any Permitted £12.40
Cheltenham Spa - Newcastle (16:11) £23.25 Advance
Total £35.65