The latching mechanism on the stock connector (JST NH series) can interfere with the usage of standard pin strips due to the additional protrusion. You may have to chop it off the cable. You won't get the latching behavior either way, though standard pin strips may be a little looser than the original connector. Might want to consider some longer than usual pin strips to get around this and provide additional engagement for better strength.
0.1" (2.54mm) connectors are also very common and will work fine over only 5 pins like you have if they're all you can find.
If you can find it, an exact matching connector will have vastly better performance in terms of pull-out resistance. Unfortunately, it seems impossible to buy this connector series in small quantity in the USA except in certain limited sizes (mostly the 3 position). Dunno if UK people are any better off.
B5P-SHF-1AA(LF)(SN) is the JST part number for the exact connector you have asked for (vertical, 5 pin header).