Well yes kettledrum but where then is what the answer should be? I do not do the Times but is this clue using 'house' or 'promontory' as part of the clue as well as the answer to be looked for? That is what confused me.
House is ARIES as Town Crier pointed out.
Almost means remove one or more letters - in this case, take the S off, leaving you ARIE (again as Town Crier said)
kettledrum pointed out one "tip" of PROMONTORY is the Y
"straddling" means wrap one set of letters around the other - A-Y-RIE
I think it's pushing it to call a nest a 'house', but if it is AERIE, then what does the straddling, tip and promontory bit of the clue give you ?
I have to say I agree with AYRIE