Invest in a decent garden fork which makes digging up weed roots a lot easier. With a fork you also take time to find any other treasures worth saving and moving elsewhere on your plot.
A good spade will help you turn the soil or mix in amendments. A watering jug will be handy if there are no hosepipes on your allotment.
Then the usuals: hat, bucket to cart away rocks or whatever, flask of tea/water.
If weeds are quite high, borrow or hire a strimmer to get them down to size.
Some netting is also needed against pigeons who love pea seedlings and other interesting green bits sticking out of the ground.
There are lots of other items you'll eventually want or need but these are good starter basics until you decide if you really like veg gardening.
Some people choose to rotovate their plots but for the most part it tends to chop up pernicious weed roots which will only multiply that many more times. Once the weeds are cut down, you can choose to spray glyphosphate (Roundup is one) to kill the roots but you must wait a few weeks for it to work.
To get a crop going, it's better if you dig up the weeds in a small patch at a time, amend the soil with compost or manure (often available through your allotment group) and start with something small like lettuce, spinach, beets or chard. Or, you could put in started tomato plants as its getting a bit late to start your own from seed.
As you work a section at a time, keep the rest of the plot covered with cardboard held down with rocks or bricks. This will suffocate any fresh weed seeds and make your job easier.
Talk to your local allotmenteers about any disease and pest problems in the area so you know what you're up against. They may even offer some leftover seedlings.