I have heard that pottery production in Britain stopped after Romans left and the shrinking cities could no longer support Roman-style ceramics production. If this is the case, what took over the roles of ceramics in food preparation, storage, etc.? Or alternatively, were there enough pre-existing pots that Brits could continue to use them without too large of a negative effect?
I am an archaeologist working in the UK and although pottery isn't my speciality (I'm a archaeobotanist) I've worked on enough sites and in the sector long enough to have an insight! Also, as a caveat, my experience is very England-centric so bear that in mind. I can confirm that pottery production in Britain absolutely did not stop following the Romans leaving. You can find pottery production happening onwards in the post-Roman period and beyond. What does change however is the style of pottery being used and it's origin.
Beginning with this latter aspect, during the Roman period you can have a good deal of imported ceramics, notably styles such as the famous 'Samian ware', which was produced in Gaul and imported into Britain, often (but not exclusively!) by members of the Roman military. This imported pottery reduces drastically in quantity following the abandonment of Britain . Interestingly, there is evidence that domestic imitation pottery manufacture of non-local styles continues for a short time- for instance there are examples of 'fake Samian' which was produced in England (not Gaul). However, beyond this imported pottery, most pottery for much of the population of Roman-Britain is regional/local manufacture. For instance, on Roman sites in Derbyshire we often find locally manufactured gray-coloured pottery called (unimaginatively) Derbyshire-ware. This practice of local manufacture absolutely continues following the Roman abandonment, however the style and function changes. Also, urban production reduces during the early Anglo-Saxon period, but by the mid Anglo-Saxon period you see a flourishing of urban pottery production. For instance, the extensive, high quality manufactures coming out of Norwich and Ipswich.
Regarding pottery styles, in a short time, certainly by the early medieval (aka Anglo-Saxon for England) period, we see a vast difference in pottery style. The migration of the Saxons and the replacement (or assimilation?- not a can of worms I'm opening here today!) of the 'native' Britons led to a vast cultural transition which is reflected in the pottery being produced. For starters, wheel-thrown pottery vanishes. We have no evidence for wheel-thrown pottery before the high medieval period (13th century I think?). We also see the use of many tools to imprint patterns on the pottery, with antler, bone and metal tools being used to produce these beautiful incisions and stamps to form gorgeous patterns. The function also changed somewhat, as the Anglo-Saxons had a very different cuisine to the Romano-British and this can be reflected in the pottery. For instance, no longer do you have nice amphora for wine/oils or mortaria for spices/herbs as these things simply aren't being consumed by the new Anglo-Saxons in the way that the Romano-British did.