If you happen to have time to read this and tell me whether I am right, or, more likely, heading in entirely the wrong direction, I would be most grateful and will then decide whether to persevere or throwin the towel!
I'm having another look at top left-hand corner this morning- I've tried to work out the answers and have come up with the following:
9 across: Sound of McCartney's instrument designed after muffling a cellar (

9 down: Creepy-crawlies found in taps (4)
2 down: It may give you a reaction (6)
3 down: Frontiers should be plain - main test, having no odd parts (6)
11 across: Most celebrated as King George renovated estate (

Answers:
BASEMENT
BUGS
ANSWER
LIMITS
GREATEST
BASEMENT: Basement for cellar; base for McC’s guitar; m for muffled??? S shared with anSwer; M shared with liMits;
BUGS: B shared with Basement and G with Greatest.
ANSWER: A from 1 across: SWALLOWS; s – basement; E - grEatest
LIMITS: L from SWALLOWS; m – basement; T – greatest
GREATEST: G – buGs; E – answer – answer; T – limiTs