TUI Engineering at Luton do work on the Max but the hangers will be fully booked out during the winter with pre-planned work.
Making a hole for a will it / will it not be delivered on xx day does not bare up the hassle. The flightline crews at larger bases, as it has been stated, are more than capable.
Additionally parking at Luton is always an issue as Biz Jets regularly overflow from FBO areas onto normal airliner stands. Currently the new 77-79 stands are being used by Harrods FBO as overflow. Wizzair UK are also doing more of their own Engineering at Luton which takes up stands close to the TUI hangers.