As Dave has said, Tubeless wires came out a long time ago and I would bet that yours are tubeless unless older tyres/wheels have been fitted.
Most plus 4's are fitted with 195 60 R15 tyres as standard, When I upgraded my tyres last year I fitted 195 65 R15 tyres. Two reasons, firstly to give me slightly greater ground clearance, yes only 10mm but on a Morgan that's a good bit extra! Especially where you are going!
Secondly It puts you into a common size tyre ie. VW polo size and gives you greater choices. For your expedition I would recommend the Michelin all season 2 tyre. It will give you the best grip in all conditions and I can see you seeing them all! It has very compliant wall structure which forms the first part of the Morgan's suspension and coupled with the slightly higher wall height it really improves the ride quality, Oh yes it truly does! The series 2 which is relatively new out has a more open tread pattern and tends not to pick stones up and then fire them out damaging the alloy wings too.
Next tyre changing. The tyre should be removed from the rear or inside rim face as the tyre is less likely to damage the sealant. Most tyre fitters will do this outer face up so be ready with the advice and insist they mount your wheel outer face down!
Balancing Wire wheels....... I can more or less guarantee they will just slap your wheel with the new tyre straight onto the balancing machine. Not only can this damage your wheel but is unlikely to produce a good balance. You need to use balancing cones for wire wheels and yep, you've guessed right, they won't have any. Can I suggest you buy a set and keep them in the car. Bet you damage a tyre or two on your adventure and will need them balancing when replacing. Should get a set for under £100 and they fit tyre balancing machines! Think the size is for 42mm cones but hopefully someone will confirm!
Tyre cost for these top of the range tyres is £112-120 per tyre. You are not racing and Avon's have a very soft structure and I think you'll shred them when the going gets tough! On the narrow body car I wonder if anyone can confirm if moving up to a 195 tyre would cause tyre/body contact. The only thing more I would suggest if not already on your car would be the fitment of brake reaction bars or cross head top stabilisation bars, again in the £100 range but will make the steering behave so much better with better precession and less fatigue. I can recommend Mulfab items.
Best of luck with your plans and don't be frightened to use the TM site for help.