Pilot went out, no problem relighting. BUT when I turn up the thermostat, the flame gets out of control and escapes from the two protective doors. It has started melting the plastic on the outside of the tank around the drain valve.

Sounds like your vent is blocked. Is there a blower on the top of this water heater?

I'm not sure. There is an opening on top of the unit and some venting over that which leads into the same venting as the furnace (the furnace is to the water heater). Then it leads out to the roof.

Well for some reason the water heater sounds like it is not drafting, or isn't getting any combustion air.

How could I go about adjusting/fixing that?

A lot of that depends. If you don't have any electrical supply to the heater, you don't have forced draft. If you do, the wires should lead to a small motor and blower. It should come on when the gas does.

Otherwise, pull the vent pipe loose from the top of the heater, and look both ways for foreign objects. There will be a twisted metal strip running to the bottom of the heater. High efficiency models may be more complicated. Check where the vent goes outside. Remove any dead birds, plastic bags, etc. you find. Except for the twisted strip and maybe a blower, there shouldn't be anything between the burner at the bottom and wherever the vent exits the house. It should have a cap to let the exhaust out but not let rain in.

No electric supply. I checked the vent over the water heater and everything seems OK, no obstructions. The vent from the water heater joins with the vent from the furnace and it comes out the roof. And the furnace is working fine. From the ground it looks unobstructed. I'll get up on the roof tomorrow for a better inspection. Many thanks.

No electric supply. I checked the vent over the water heater and everything seems ok, no obstructions. The vent from the water heater joins with the vent from the furnace and it comes out the roof. And the furnace is working fine. From the ground it looks unobstructed. I'll get up on the roof tomorrow for a better inspection. Many thanks.
Also check the core of the water heater it may have sooted up. Also fresh air comes in from the bottom of the water heater, a round hole just under the burner, this could be blocked with rust and stuff, look for other openings or screens if this is what's called a sealed combustion water heater.

I have a definitive answer: My water heater imploded. Thanks to all of you for help, it is much appreciated.