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skinny
What you are looking for is a 'wireless access point', not wifi booster/range extender/repeater which are about extending well, wirelessly, rather than via Ethernet. (Although the terms seem to get confused, especially with powerline networking... )
Very cheapest way of doing it is to use an old wifi router and put it into 'bridge'/'client' /'AP' mode. This may not be possible on BT home routers though. Should be easy to find an old Netgear/Linksys/TP-Link etc for a couple of quid though especially if you are OK with older wireless N.
If not something like this does wireless AC: https://www.ebuyer.com/748436-tp-link-ap300-ac1200-wireless-gigabit-access-point-ap300 for about fifty quid.