Is it safe to daisy chain extensions cords?
Extension cords are more dangerous than direct connections. More extension cords plugged in series are yet more dangerous. However, life is dangerous, and we all accept some danger, in driving or cycling on the roads for instance. The regulations around extension leads and driving try to reduce that danger to some low level that we accept as reasonable.
The fuse used in UK plugs generally protects the wiring against overheating due to a short-circuit, if it's properly sized. Too often, we see a standard 13 A fuse used with undersized wiring. Note that a fuse will not generally protect against a low level of overload, a fuse will typically carry 200% of rated current for long enough for an under-sized cable to overheat.
However the fuse also provides one point of danger. Strained springs contacting the fuse could cause a high resistance, which could overheat and start a fire. Other points of potential high resistance are loose screws where the wires enter the terminals, and loose springs in the sockets connecting to the plug pins. This last one is exacerbated if the plugs are not fully in.
Adding one fused extension lead to a circuit adds another 8 (eight) series potentially loose connections to the circuit, many of which could start a fire. Every further extension lead connected adds another 8 connections.
Every extra extension lead added offers to opportunity to bury the cable underneath a carelessly tossed cushion or pile of clothing, or maybe route it tidily under a rug. This thermally insulates the cable, making it reach a far higher temperature than would be the case in free air, and providing fuel to burn should the cable catch fire.
Extension leads are also trip hazards.
Many extension leads confuse the route. If you want to quickly plug in a fan heater, which is the best socket to use?
In your case, a soldering iron and a couple of PCs comprises a load so far below the rated current that it's likely to be safe. Keep the cable route clear of walkways, and clear of clothes. Check all the plugs from time to time to make sure they are fully inserted.
If your situation changes, more people, piles of clothes, need for fan heaters, then re-assess the dangers.