There are several ways of dating cave fills, but all have their drawbacks and most are expensive (usually hundreds of ?? per sample), and in general more than one sample is normally dated to make sure they are reliable. So, if you want go ahead, make sure you have something datable and a good reason for doing it! If you have a really good reason for dating, then you may be able to get a tame scientist to do the work for you.
1. Dating of any associated speleothem calcite is relatively quick and cheap these days using U-Th methods. Assuming good clean, non recrystallised calcite with no detrital contamination and usuable amounts of U, this is OK back to c. 500 ka. This is probably the best method for getting a reliable date, and only requires a very small sample (milligrams will normally do). It has been used extensively in the UK, for example a thick stal layer in the bank of gravel by the Bridge in GB has been dated to c. 50 ka, but this dates the timing of stal deposition, not the gravel fill per se. Several labs in the UK do this (Bristol, NIGL at the British Geological Survey). Older speleothems can be dated by the U-Pb method, but this relies on decent U concentrations and low 'common' Pb values.
2. Palaeomagnetic dating. This can be used on fine grained muds, but will only tell you if the sediments are normal or reversed polarity. The last reversal was c. 780ka. There are several caves in the UK which are old enough have reversed polarity sediments (GB for example), but most are not. It also requires quiet deposition in still water to enable the Fe-rich clay particles to orient themselves to the prevailing magnetic field, rather than water flow. There are several UK labs that can do this (eg Plymouth Uni). Under ideal conditions it may be possible to determine the secular variation, but this is unlikely. Worth doing if you think the sediment is old enough, as this is beyond the range of U series dating.
3. Cosmogenic isotopes (10Be/26Al). This relies on quartz that has been exposed on the surface before burial in the cave, as the isotope ratio will vary with time on burial. However, this is not cheap, and is best suited to older samples. It has been used in the USA to date the upper levels of Mammoth Cave back several million years, but has not been used much in the UK.
4. Palynology - this won't give an age unless you have some really good indicator pollen species (assuming you get any), and even then is open to interpretation.
5. OSL. This relies on quartz in the sediment being exposed to sunlight prior to emplacement in the cave, so has limited applications for cave sediemnts, unless they are unequivocally derived from the surface. Samples are about ?600 a pop.
6. Radiocarbon. Needs organic carbon, and prone to contamination from other sources of carbon (ie limestone). Only dates back to c. 40ka, so not really used for interior cave deposits.
7. Fauna. If you find any bones, they may provide a rough date (mammal assemblage zones), eg hippo is normally associated with Marine Isotope Stage 5e at c. 120 ka. You'll need Andy Currant at the BM!
Best way to date a cave fill is to have some speleothem encrusted mammal bones in it, ideally very old early hominid remains in it, adjacent to laminated clay intercalated with peat.... That way you'll get an archaeologist to do the dating for you!
I would try U-series on the calcite in the first instance. Its not cheap, but there are grants available, but you'll need a good proposal to get the funding.
Andy