The Geographers' A-Z Map Company (who do the famous London A-Z street atlas) also do similar roadmaps for every UK county and city, including Dorset (county) and Bristol (the city of your two stated destinations).
You can buy all their products online , and at large stationery shops, for example WH Smith, you will find the relevant local maps.
The South West England and South Wales
A-Z Road Map covers Bridport and Bristol on one map, although you might prefer to use road signs to get into the coverage of the larger-scale 50 miles around Bristol map.
That said I would heartily recommend that one of you downloads the Waze app on their smartphone and uses that for navigation.
Waze has the google map data and also, crucially for UK driving, has access to crowdsourced live traffic data.
You really don't want to find yourself on the M5 on one of its bad days...
travel.stackexchange.com/users/3884/AakashM
The Ordnance Survey do road maps, and the one you'd need is sheet 7 - South West England .
Saying that, for actual navigation I'd always recommend a traditional road atlas (combined with, as AakashM says, a traffic-watching GPS to avoid the snarl-ups on the M5).
You should be able to get one in any decent bookshop, or in most petrol stations or motorway services.
I'd got for one of the "concise" ones that are a smaller size, as the big A3 size ones can be rather unwieldy.
Something like this one from the AA, for example.
travel.stackexchange.com/users/18319/Nick C
If you're sure you want paper maps, it's probably best to just get a UK road atlas.
The UK isn't a very big country so they're not very expensive.
travel.stackexchange.com/users/11491/David Richerby
Paper road atlases designed for exactly this are usually pretty widely and cheaply available in the UK from stationers such as WH Smith or Waterstone's, or any of a number of independent local retailers.
You could probably buy one in advance from your favourite online retailer to be delivered to your UK destination, ready for you.
A nationwide atlas is likely to be perfectly workable and not that expensive (under GBP 10); I've usually only bothered with county-specific atlases (note, as another poster said, that the sites you're looking for are in Bristol, not Dorset) where I need fine detail of residential areas.
All that said,
I wouldn't buy one and haven't in years.
Waze is an excellent free GPS navigation application with crowdsourced traffic data and more up-to-date maps which will make your journey much smoother.
If you're concerned about data charges, Google Maps is also excellent for satnav and can be used completely offline if you download maps in advance - it doesn't take much device space for a really quite large area, and I traveled round northern Scotland that way last year with almost no signal but excellent navigation.
travel.stackexchange.com/users/74772/eftpotrm
Most bookshops sell paper maps - Waterstones is the big national chain, but local bookshops will sell them too.
You can also pick up 2-3 year old road atlases in many bargain bookshops (e.g. The Works ) for about £2.
It's probably cheaper than buying them online, given you'd have to pay postage too.
If you head to the local tourist information centre, they often have free maps within leaflets for either specific attractions, or a group of local places.
Depending on what you're after, they may be sufficient in some cases.
They certainly would be as a supplement to a national road atlas.
travel.stackexchange.com/users/38044/Algy Taylor
Just buy one of these, https://www.amazon.co.uk/2018-Collins-Road-Atlas-Britain/dp/0008214557
I think those are still widely available in the UK.
bookshops at the airport (when you arrive)?
at petrol stations
en route?
certainly at large bookstores in cities.
I much prefer "real, old-fashioned" paper folding maps, https://www.amazon.co.uk/Explorer-Purbeck-Dorchester-Weymouth-Swanage/dp/0319242544
but they're harder to find.
(Well, trivially available online - if you have time just buy one from amazon .co.uk
and they'll post it to you in Oz.)
There's actually a famous map shop in London, if you're in London you should go ..
stanfords.co.uk‎
travel.stackexchange.com/users/19233/Fattie
An option no one else has mentioned is to pop into a local Halfords , which is your one stop car/bike/cycle parts and motoring-related-tat superstore chain - the nearest one to Bridport is in Dorchester.
travel.stackexchange.com/users/29788/Moo
As an ex-Brit I'd vote for buying a cheap road atlas from a service station (or Halfords) when you land.
But, another option
if you're in the Perth Metro W.A. and desperate to plan ahead is to try here:-
http://www.chartandmapshop.com.au
It's a great shop to just browse in and should have what you're looking for.
travel.stackexchange.com/users/74797/mcottle
I'm coming from Australia to visit my Dad in Bridport (UK) in Dorset.   We want to drive to Bristol to see the  Brunel Museum  and we want to direct ourselves to see the  Clifton Suspension Bridge  on the way.   To plan out our driving journey, we'd like to get a paper map.   My question is:  How to get driving (paper) maps in Dorset?
How to get driving (paper) maps in the county of Dorset?
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Your best option is probably to buy a roadmap for Dorset or the UK, which should be available at any decent bookshop, petrol station or motorway services, as well as Halfords and online. 2-3 year old road atlases are often available very cheaply at bargain bookshops. You could combine this with GPS to avoid traffic on the M5. The A-Z UK atlas seems to be a very popular choice amongst many drivers who have to navigate the UK. They are avaible in an abundance of places such as, WH-Smith, Halfords, Garage sales and even online. Aside from that you can also find more traditional paper maps such as "sheet  7 - South West England" in bookstores such as Water Stones or your local book store. 
The A-Z UK atlas series seems to be a veru popular book for navigating the UK using paper maps. They are avaialbe in multiple stores such as WH-Smith, Halfords, Standard bookshops and even bargain sales. They are also available online. A more traditional form of paper map can be found in most bookshops and the specific one you would look for is sheet  7 - South West England paper map. You may find maps like this a Waterstones as well as smaller local bookshops.