My question is how many fibres are there in the optic bundle they're running up from the exchange to the 32 way tray splitter?
In one sense it doesn't matter how many go from the exchange to the splitter. What matters is that you have one 32-way splitter fed by "all" those fibres.
A house doesn't get service from the fibres coming from the exchange - it gets service from the fibres coming out of the splitter.
To be added to the service, you need there to be a spare port on the splitter - and we don't really know how many of the 32 ports BT consider to be a usable amount.
Unless, of course, it isn't a splitter, and is really a fibre DP. Are you sure it is a splitter?
The cable says Blown Fibre Tubing Mark3(2.7/4.0mm) and on yellow tags it says Fibre capacity: DAAIPQQ-DAAIPQS
BT's architecture would normally put the BFT from a fibre DP out to a manifold located in a chamber or on a pole near the property; those manifolds are normally for 7 or 12 fibres, and are used to convert BFT into the final drop into the house.
You'd guess, then, that BFT tends to come with 7 or 12 tubes.
Are you sure the BFT holds fibre coming into the trays from the exchange, rather than fibre going out to the homes?
The general architecture used by BT has multi-strand fibre cables out as far as the DP, and the cables tend to have a number of strands that are a multiple of 12. We've certainly seen mention of 48 and 96-fibre counts. The biggest cables can have up to a 288 fibre count.
Of course, some of these will be dimensioned as spares in case of breakages in the decades to come, and some will be for future point-to-point use (eg G.fast nodes, or business use).