Services on A1 south

No, Grantham (Downtown) services doesn’t have dedicated caravan spaces.  We use it often, last time in May, there are usually spaces around the edge of the HGV area or with the lorries.  Never failed to find a space there somewhere. . 

Caravans are dtrected into the car park, we used to use it every year and never had any problem finding space in that area occupying two spaces. if you keep to the right on entering car park usually plenty of space to park. Only 15 dedicated spaces for HGV, usually more space in car park, everyone wants to park near services entrance.

That will only work if your whole outfit is no longer than 2 cars

Worked for me and other caravans just overhang the bays, plenty of room between the rows.

That’s what we did. It’s not a busy car park there’s usually plenty of space.

Regarding heading for East Anglia from the A1, I try to avoid the horrible roundabout at Newark. I find it’s much better to go south to the next junction signposted “Coddington” and leave the A1 there it’s not far from there to the A17

Re Newark roundabout, will try that when we go to Cromer in November without the van. Only used it solo, and that is bad, must be a nightmare with a van.  Usually go once a year and stop at Friendly Farmer for lunch both ways, but looks like it’s permanently closed, so looking for somewhere else.

Swavesey Services on the A14 has a dedicated caravan area with the bays set on a diagonal and allow units to drive in and then drive out. No reversing required 

I realise it's too late to answer the OP's query, and I hope you had a good trip, but out of interest - we towed our 'van down the A1 from near Berwick upon Tweed on Tuesday this week, intending to pause at Wetherby.
The road was moving very nicely all the way, but Wetherby services was jammed solid right back to the entrance road, so we wickedly did a U turn while nothing else was moving, and headed on down.
On earlier visits, yes, as long as it's not jammed up, it's been a decent place to stop.

(Irrelevant here where we did end up, as we came home via the M18/M1,)

This is us trying to fit into the caravan bays  at Baldock, and the necessary solution.    
As you can see, the front of the car is already over the front line, and half the caravan is still in the road!   The bays are the length of 2 cars, and only very slightly wider than the caravan.

Looking at the photo of Grantham services, I would say we would have had the same problem.

Note also the less than ideal angling of the bays.....they are angled to the left, while the exit road is off to the right.  Not possible to exit to left and drive round as they had fenced off some building work going on there and partly blocked the road.

We use the services at A1(M) junction 17 near Peterborough. However it often gets very full at popular times. If that's the case we just cross our legs and keep going. Welcome to Suffolk. No motorways here!