Three issues affect the usability of these wheels; #1), the over locknut width of the wheel and your Dawes’ rear drop out spacing. Depending on the year of manufacture, the Dawes may have to have its drop out width adjusted to accept the usually wider freehubs. If you can’t measure the hub width, specs are usually available on line.
#2), the brakes on the bike need to have at least 4mm of extra area to lower the brake pads to. Lower, in this case, “lower” means closer towards the wheel’s axle.
The most common Dawes model that I have seen in the US is the Galaxy model of theSeventies, comparable to a Raleigh Super Course. These were, IIRC, supplied with Weinmann Vanquer Centerpull brakes. I would bet money that they have the necessary extra 4 mm needed to accommodate the 700C wheels.
#3), the Dawes’ derailleur must have adequate speed capacity to work with the freehub’s increased number of cogs. The derailleur could be changed to one that has adequate capacity for this wheel. A derailleur change usually calls for a new cable inner, at the least.

The advantages for changing the wheels is really for #1) a broader variety of tire options. There are fewer models available for 27” wheels, but there are some pretty good options available from Schwalbe and Panaracer in 27”.
#2) The offerings for freewheel combinations are also dwindling, these days. If t you needed to replace the freewheel on the 27” wheel, you might not find a suitable replacement that would align with your needs. Freehubs are prevalent now, so there are more options commercially. There are people like Pastor Bob who might have more guidance on available options for freewheels.

Hopefully, this gives you enough iformation as to whether this swapping of wheels is worth your effort and time.