Stopping cordials fermenting naturally is always a bit difficult without some chemical inhibitors, vigorous pasteurisation or freezing. Wild yeast and bacteria can be difficult to eliminate but some methods produce better results that others.
As noted, pasteurisation is a good "catch all" method if carried out VIGOROUSLY. I don't mean boiling vigorously I mean being certain to boil and quickly transfer to the storage bottle. In this respect glass bottles are best because one can sterilise them easily by filling the bottle with boiling water. Fill the bottles with boiling water (and if with flip tops - cap them), then leave for a few minutes. Sterilise a funnel with boiling water (by pouring it over). Meanwhile bring the cordial to a boil (or at least to approx 85C for a few minutes), then empty the bottles of water, insert the sterilised funnel and fill with cordial. Cap immediately and even "rinse/wash" the cap/top of the bottle with more boiled water to clear any remaining drops/film of cordial. Leave to cool before storing in a fridge - although with this method it should be possible to store the cordial at room temperature.
Another possible method is to fill a plastic bottle and put the bottle in the microwave to pasteurise it. This needs observation because the aim is to get the cordial to about 70C for (say) 30 seconds to pasteurise effectively - but not to boil it! That can be difficult to achieve.
Some people don't like boiling cordials with a delicate flavour - like elder-flower cordial. In such cases the cordial can be pasteurised at 65C for a minute or two and then (for prolonged storage as opposed to quick use) one can add half a metabisulphite (campden) tablet to each bottle. Alternatively one can add potassium sorbate - which some consider preferable and "more natural". Both these compound inhibit yeast and bacterial growth and so will slow down (or stop) further natural fermentation.
If none of these methods is preferred then one can make cordial and store it in plastic bottles in the freezer. That will keep the cordial fresh and fragrant and all that is needed is defrosting before use. A friend recommends this method for elder-flower and uses an ice cube tray to make "cordial cubes" - once frozen the cubes are stored "en masse" in a freezer bag. Take some from the freezer as required and add to water or soda for a cool instant drink. No preservatives or boiling required.
Fruit cordials are essentially "very watery fruit jam". The most important thing is to liquefy the fruit and dilute it sufficiently to allow it to be efficiently filtered/strained to remove the excess pulp/stones/pips/leaves/stalks, etc. Once the fruit is sufficiently pulped and strained then the resulting must can be brought to the boil, sweetened with sugar, bottled and stored. Blending or mashing followed by filtration are IMHO the important steps to extract the maximum fruit essence and produce a smooth drink.