Would the same production be possible today? I don't think they are comparable. We no longer build fighters out of wood and canvas. Today we could absolutely out-produce 1940s Britain for Hurricanes and simple Spitfires. Could we produce as many modern fighters as 1940s Britain produced old-fashioned ones? I simply cannot answer that question, I'm afraid.
As to your other question, of how it was done in WW2... put simply: with a great deal of difficulty and effort.
It wasn't smooth sailing, especially in the early years. There were two main British fighter aircraft early in WW2: the Hurricane and the Spitfire.
The Hurricane had a simple construction and was easy to put together. One of the main materials was wood. Another was canvas. It used a standard engine. Hurricanes could be put together very quickly.
Spitfires, however, were all metal: which was a recent development. They had a rounded wing instead of straight: which was more difficult to manufacture.
Compared to modern fighters, however, both were very, very simple. The manufacturing process would be child's play for modern equipment. But (to put this into context), 1938 was only 30 years since Henry Ford rolled the Model T off the lines. Mass production was still quite a new idea.
For any large company, the production targets would be a challenge. But Supermarine (the company who designed the Spitfire) was a small company. They had to subcontract out to cope with the size of the orders that they were getting from the British government. The process did not go smoothly. The first technical drawings which Supermarine supplied to other companies were not of the required quality, and parts which had been made to the specifications sometimes just simply wouldn't fit together when they were assembled.
The RAF had planned to replace all their older fighters with Spitfires and Hurricanes before 1939. But the aircraft simply hadn't arrived in the numbers needed. Britain entered the war with many outdated fighters. Some biplanes were even still in use as combat aircraft up until the early days of the Battle of Britain (where they were outclassed and quickly withdrawn after heavy losses).
Precious few Spitfires existed.
It was a major crisis, which took a great deal of ironing out. And that didn't happen until after the Battle of Britain was over. Though everyone associates the Spitfire with the battle, it was the Hurricane that was the most common British fighter by far.
The British government had built a factory specifically to build Spitfires, at Castle Bromwich. The land was purchased in 1936. In 1939, the project had doubled in cost and was not yet completed. Production only began in early June 1940, after the factory was taken over by Vickers (the parent company of Supermarine). It was years behind schedule.
In May 1940, Winston Churchill appointed Lord Beaverbrook as Minister for Aircraft Production. This was a new post, designed specifically to handle the production problems which had become critical. He made a dramatic impact.
Many of the improvements to production were made before Beaverbrook took charge. But it was Beaverbrook who brought together all the strands of production and made them co-operate effectively.
He also took charge of the Civilian Repair Organisation, who took badly-damaged aircraft and repaired them, or cannibalised them for parts. This was a highly efficient operation, and aided the supply greatly. The Germans drastically underestimated how many aircraft could be repaired.
But the production problems continued through the war. To quote directly from my source:
Throughout 1942... not enough spitfires were coming off the production lines... At Castle Bromwich, now finally reaching its potential production levels... [lack of skilled labour] led to quality control issues... To keep up production numbers, large numbers of hydraulic power press tools had been put into service, starting in 1941... but changing tooling required major effort.
It goes on:
The British economy.. was creaking under the strain. Part of the answer was to bring more women into the factories. By 1941.. the workforce had become 30% female.
In short, the British managed to produce many aircraft simply by building enough factories and by having a very efficient repair service. But the factories were very troublesome in their early years of operation, and it took a long time to sort out the processes.
I've done my best to summarise my source, but there is literally an entire book on the subject. I recommend it.
"The decisive Duel: Spitfire vs 109" - David Isby.