That's just up the coast. I presume you meant to type Whitby.
I did St Bees to Robins Hoods Bay back in the mid 90's, with two mates, one of whom I had never cycled with before and he turned out to be problematically slow. We used the route shown in MBUK back then. We planned to go up Scafell Pike too, but were beaten back by foul weather. We did it in April, which being all soft southerners we thought it would be lovely and sunny...cue me shouldering the bike and fighting through waste deep snow on the climb from Kirkby Stephen to Tan Hill.
I remember getting a round in, in an inn in Kirkby Stephen
"That'll be £3.06 please."
"Aye, sorry, we've 'ad to put us prices up"
"No, I was expecting at least £6"
No support, just a carrier mountable rucksack each, stopping in Youth Hostels and my In Laws B&B
Route was St Bees to Buttermere, Buttermere to Town End (near Ambleside/windermere), over via Shap to Kirkby Stephen then to Grinton Lodge, Richmond, Ingleby Arncliffe, moors over to Whitby/Robin Hoods (that bit was mainly road as time was running out, shame to waste the moortops)