Katie Boulter won her first match since becoming British number one to move into round two of the Nottingham Open.
Harriet Dart and Jodie Burrage were also first-round winners in the build-up tournament before July's Wimbledon.
Dart beat Canada's Rebecca Marino 6-4 6-2 and Burrage gained a hard-fought 7-6 (7-5) 3-6 7-6 (7-4) win over Tereza Martincova of the Czech Republic.
Boulter is 126th in the world rankings, with 128th-ranked Raducanu currently out of action after hand and ankle surgery.
Burrage, Swan and Dart, 131st, 134th and 143rd in the rankings respectively, could overtake Boulter next week.
Heather Watson, who won her first-round match on Monday, is 195th and a former British number one and could also regain top spot, but would need to reach the final to have a chance to do so.
Boulter became the 23rd woman to become British number one since the rankings began in 1975 and needs to reach at least the quarter-finals in Nottingham to have a chance to stay there.
In a post on Instagram, she wrote: "Been mulling over whether I was going to post this today.
"It's not my highest ranking nor my best career moment however someone reminded me today that this little girl would be proud if I told her she would be British number 1 one day.
"Whether it be for one minute, one day or one year, it's not my biggest goal but shows I'm heading in the right direction. Now I've got work to do as my main goals are pending."
Against world number 420 Appleton, Boulter dropped only four games but in the next round she could play French Open semi-finalist Beatriz Haddad Maia of Brazil - if she defeats Ukrainian lucky loser Daria Snigur.
Murray in action after first title in Britain since 2016
Three-time Grand Slam champion Andy Murray will also be in action in the men's Challenger event, fresh from his win at Surbiton last week.
Murray claimed his first title on home soil since Wimbledon in 2016 and is looking to build on that before the grass-court major starts on 3 July.
The 36-year-old took part in a light practice on Monday with coach Ivan Lendl by his side.
Murray plays Belgian qualifier Joris de Loore on Tuesday and you will be able to watch live coverage of that match on BBC Red Button, the BBC Sport website and app.
Britain's Liam Broady, Arthur Fery, Jan Choinski, Ryan Peniston and George Loffhagen are all scheduled to play on Tuesday as well.
Play at the Nottingham Tennis Centre began on time at 11:00 BST, despite a series of knife and van attacks in the city in the early hours of Tuesday in which three people were killed.