I am also finding it hard to find quotes. May be able to via Admiral (my current insurer), but again I've not investigated the X, only S for an equivalent estimate.....I am also in London (parking on a drive vs garaged) - do you mind me asking the name of the broker you have contacted?. I contacted my DS who suggested LV and Home & Legacy as two that might work.
Here is my experience - Model X P100D, price £140k, parking in Marlow on driveway, husband and wife in 40s with no convictions and 9 years plus no-claims, late Dec delivery (hopefully!):
My multi-car policy with Aviva is up on 14th Dec anyway, so was phoning around to potentially switch anyway.
NFU - no longer taking on Tesla, will maybe renew for existing owners.
Churchill - quoted about £900 using an S P85D as a guide, with a £1350 excess
Adrian Flux - £1575 for a bare bones policy
Direct Line - wouldn't touch it, even their "premier" service. Rumours on FB are that they are insuring Teslas though with a 5% discount for AP, may be my value that scared them.
LV - not insuring cars over £100k
H&L - Big on Teslas, turned me down because of the value and the fact I occasionally park at T5 short stay :-(
Aviva - My existing insurer, not insuring Tesla.
Admiral - sorted, got a good multi-car deal to include my existing cars that matched what I was paying with Aviva, and worked out about the same as Churchill but with less excess and doing my other cars. They had no real problem quoting on the X, but did take an afternoon to put it onto their systems. We'll see what happens when I phone them up next month to put the X on when I get my delivery date confirmed, but they assure me they will have no problems...
The bottom line is, owners are having luck with all of the above bar NFU, however the value of a fully-loaded X puts you sadly into tricky territory with a lot of them. It also seems to depend on who picks up the phone!