Train/Drive/Train combo from Edinburgh to York to see Lindisfarne en route?

Have you already checked the safe crossing times for the day you want to travel there?

@CJean yes, the hide tide window that day is from 9:40am until 1:20pm (and then again late in the evening). I expect we’d have a more leisurely start if we visited Alnwick first and then went to Lindisfarne however as I’d expect all the tour buses that day will schedule their Lindisfarne stop for the afternoon, I’m more inclined to have an early start to get to Lindisfarne before the tide comes in (we’d probably aim to be there by 9:15am).

With that, I’ve just had a look and there don’t seem to be any car rentals in Berwick so may very well be that we have to drive from Edinburgh or find another stop on the train route that has a rental facility in town.

If you’re into Abbeys and had the time consider stopping at the border abbeys (Jedburgh, Melrose, Dryburgh, Kelso). They are close together, but probably out of the way if picking up the rental car in Berwick-upon-Tweed. If picking up the rental at Edinburgh airport, it’s a 2 hour drive to Alnwick, but there may be a large drop off fee dropping off the car in England. You’d need to check how much the fee would be. A dummy booking can give you that amount. There is an Enterprise car rental location in Ashington, just outside of Alnwick. There are a number of major rental companies in Newcastle upon Tyne; Enterprise, National, and Europcar at the train station.

We stayed a couple nights in Alnwick and had plenty of time to visit Holy Island. You can check the tide schedule online. We visited the island and left before the tide came in. We then visited Bamburgh castle and returned to Holy Island at high tide.

Alnwick castle will take a few hours to visit if you’ve never been to it previously. Since you’re headed to York afterwards, returning the rental car to Newcastle upon Tyne probably makes more sense.

Newcastle is actually not that bad to drive in. Most of the wider city is 20th C so more developed with car travel in mind than some places. Obviously it’s always best to avoid the centre of any city because you’ll come up against bus gates, no entries and all of that fun stuff, but it shouldn’t be necessary to do that in order to return a hire car.

To Central Station in Newcastle for return rental you pretty much come straight off the Central Motorway in Newcastle - it has a road number but everyone just calls it the Central Motorway. It is not really difficult.
There may also be the service bus from Berwick station to Lindisfarne running that day.
Another way to get to Lindisfarne is to go to Seahouses and use Billy Shiels High Tide boat service.I
The Castle is usually only open at Low Tide. Tour buses are not a big issue d there are not huge fleets of buses going to the island. And anyway in the village people very soon dissipate. Let alone on the way to the Castle or out in the Nature Reserve/beaches and sand dunes at the north end of the Island which are always quiet .At least a tour bus is 40 or 50 less cars so less cars taking up the parking spaces.
There is no conventional car rental at Berwick although Enterprise supposedly have a car club there.
Another option would be a taxi from Berwick.
There is actually car rental at Edinburgh Waverley station. It may be worth doing that and facing driving out of the City Centre rather than going out to the Airport then taking the Southern Ring road out to the A1- in pure time terms.
In many ways a guided day trip is a much more efficient way to do this itinerary, they take different routes out and back so you also get to see some of the Scottish Borders.
Then just go straight from Edinburgh to York.

I did this last September. After our time in Edinburgh, we picked up a rental car at Edinburgh airport to avoid driving in the city. We stopped at Rosslyn Chapel (was used for the end of the Da Vinci Code https://www.rosslynchapel.com/) on the way to Alnwick. We stayed there for 3 nights while we did Lindisfarne (abbey and castle) and Bamburgh Castle (locals say there are magnificent views of the ocean and the islands from both but we couldn't see the water (which became a running joke). Alnwick Castle was a total surprise and very good (movies sets for Harry Potter and 2 Downton Abbey Christmas specials). Look for the stuffed rats in the state rooms (ask a guide for an explanation). The entry ticket is good for more than 1 day so we did the state rooms at the end of the day we went to Lindisfarne and then went back for a morning before leaving Alnwick.

We then went on to Hardians Wall before returning the car at the Newcastle Airport. We then used the train to go to Durham and then York. Train trips were easy and easy to skip Durham to go straight to York.

Getting to Lindisfarne and Bamburgh without the car would be difficult. Also the train station is not in Alnwick, but in Alnwick Mouth.

In many ways a guided day trip is a much more efficient way to do this itinerary, they take different routes out and back so you also get to see some of the Scottish Borders.
Then just go straight from Edinburgh to York.

I totally agree with what isn31c said.

Having looked at the map, we’re now considering doing it ourselves and then continuing onto York to avoid backtracking.

I would not say that doing a day tour out of Edinburgh is "backtracking".
The fact that you return to Edinburgh at the end of the day trip is a good thing, as you can then use the train to go south to York the next day.
Your plan is simplified and made less stressful by not renting the car, in my opinion.

P.S. On your train trip from Edinburgh to York, consider getting off at Durham for a look at Durham Cathedral.
Then hop back on the train and continue on to York.
Or if you have lots of time on this trip, consider staying one night in Durham.