Can you please tell me which travel system you have and your baby's current height, weight and age?
It is possible you can keep using the seat you have safely. They are not always outgrown as quickly as people think.
If you book your taxis in advance, call around to see if you can find a firm which has child seats. Some may have longer lasting rear facing seats - they are much more common these days. Though you might prefer to stick to a seat you know the history of.
It might be possible to buy a new one which is a similar type which will last a good few months longer. There is a Britax one on offer currently under £50 which is good, and compatible with most travel systems.
If baby is over 15 months, there are some forward facing folding seats which fit using top tether and isofix which are much better than nothing, although it is best to keep babies rear facing as long as possible. So personally I would try to keep baby in the rear facing travel system seat as long as they safely fit, then I would swap to one of these folding seats - an R129 approved one like the Kinderkraft, MyBabiie or Cosatto, not the fabric ones which are sold online (those are not crash tested or legally approved). It is well worth spending the extra £10 on one with isofix rather than seatbelt fitting.
Legally you do not need to use a car seat in a taxi, although do be aware if the driver were to crash, it would be high risk for an unrestrained child. In private cars, children must use an approved child seat up to 135cm height.
Some people use black cabs and park the pushchair facing backwards in the wheelchair space, this is a reasonable compromise at city speeds, though a pushchair is NOT crash tested, so be aware that you do not know what would happen in an accident, and it cannot offer the same protection as a car seat.
If you ever travel in a taxi with a baby on your lap, wear your own seatbelt and hold baby (under 3 years) in your arms but do NOT put your seatbelt over them at all. In an accident, that has a high chance of causing fatal injury. If they are over 3 years old, put them on their own seat wearing the seatbelt as an adult would - don't tuck it under their arm. It won't fit properly, but it is better than nothing. Use a backward facing seat if you have the option.
This is basically the run down:
Best > rear facing approved child seat
If not practical > forward facing approved child seat
If not available > own restraint (e.g. seatbelt) is always better than none, backward facing seating is preferable.
Never share a seatbelt with a baby or child.