The best time to pollard and lop a sycamore tree would be more towards late autumn winter when its lost its leaves and stopped frowing for the the winter. Its worth baring in mind syamore trees are heavy bleeders of sap spring-summer which could introduce disease from insects or fungai, so I wouldn’t do anymore than light pruning in these months.
A mate is a tree surgen and runs SWA in south Wales, which is probably too far for you. However for referance he charges £175-£500 half day to full days work for pollarding and the same range for lopping a tree, with a minimum charge of £395.
From memory when Ive gone to collect the logs for firewood they measure the girth/crown/height to calculate a base price.
A tree surgeon also offers the service of mulching the branches which can be used around garden area’s to surpress weeds and keep soil moist ?(they often do this on site but charge you if you want it). It is stll more cost effective than buying bags of the stuff. They will most often leave you with the logs if you have a log. burner or they take them and sell them on.
£4k sounds like he was charging you to fell a big old oak tree in an unsafe location!!