Now I've done it a couple more times, I thought I'd post back, just in case anyone ever asks a similar question. I'm only going occasionally, as we're still mostly working from home even now, so it's only three trips in total.
As I mentioned, the CycleStreets route was a bit congested, especially around the crossing of Regent Street. I've actually found it a lot better to keep a bit north of that, more in line with the suggestion from grams. So I've used the Tavistock Place - Howland Street cycle route, in both directions. Westbound, Great Portland Street and Margaret Street take me onto New Bond Street, which is a good run south to Bruton Street, Berkeley Square and Curzon Street, then some small back roads. Only seems to add a couple of hundred yards over the CycleStreets route, and quite a decent route overall. The first half I knew already from trips to other places.
Coming back, I tried South Audley Street and Adams Row to get onto Davies Street and Marylebone Lane, but found it a bit plagued with lights every hundred yards, some of which apparently failed to respond to bikes. I think I'm going to see whether it's better just to use Hay's Mews to avoid some of the congestion in Berkeley Square getting onto Davies Street, and I've an idea that Wimpole Street might avoid a recalcitrant set of lights at the top of Marylebone Street that don't even seem to respond to cars at the moment. The A5204 is then a good run to Chenies Street and Torrington Place again.
I can do the westbound route in about 20 minutes, but the wind-assisted return is taking five minutes longer, which gives you an idea how much stop-start you get from too many sets of lights! Anyway, that's my answer for now. Thanks for all the suggestions.